Proof of Corollary 1

Summary of the proof

All three residues

Simplify[3Residue[Residue[Residue[f1[s,w1,w2],{s,1}],{w2,0}],{w1,0}]-3Residue[Residue[Residue[f2[s,w1,w2],{s,1}],{w2,1}],{w1,0}]+Residue[Residue[Residue[f3[s,w1,w2],{s,1}],{w2,1}],{w1,1}]]

Summary of the constants

f1[s_,w1_,w2_]:=X^((w1+2w2+3s)/3)/w1/w2/s*Zeta[s]^3*Zeta[w1+w2+1]*Zeta[w2+1]*A1[s,w1,w2];
f2[s_,w1_,w2_]:=X^((w1+2w2+s)/3)/w1/w2/s*Zeta[s]^3*Zeta[w1+w2+1-s]*Zeta[w2+1-s]*A2[s,w1,w2];
f3[s_,w1_,w2_]:=X^((w1+w2+s)/3)/w1/w2/s*Zeta[s]^3*Zeta[w1+1-s]*Zeta[w2+1-s]*A3[s,w1,w2];

Derivative[0,0,1][A1][1,0,0]:=A1[1,0,0]*Derivative[0,0,1][A111][1,0,0];
Derivative[0,1,0][A1][1,0,0]:=A1[1,0,0]*Derivative[0,1,0][A111][1,0,0];
Derivative[1,0,0][A1][1,0,0]:=A1[1,0,0]*Derivative[1,0,0][A111][1,0,0];
Derivative[0,1,1][A1][1,0,0]:=A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[0,1,0][A111][1,0,0]+A1[1,0,0] Derivative[0,1,1][A111][1,0,0];
Derivative[1,1,1][A1][1,0,0]:=A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[1,0,0][A111][1,0,0]+A1[1,0,0] Derivative[0,1,1][A111][1,0,0] Derivative[1,0,0][A111][1,0,0]+A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[1,0,1][A111][1,0,0]+A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[1,1,0][A111][1,0,0]+A1[1,0,0] Derivative[1,1,1][A111][1,0,0];
Derivative[0,0,2][A1][1,0,0]:=A1[1,0,0] ((Derivative[0,0,1][A111][1,0,0])^2)+A1[1,0,0] Derivative[0,0,2][A111][1,0,0];
Derivative[0,2,0][A1][1,0,0]:=A1[1,0,0] ((Derivative[0,1,0][A111][1,0,0])^2)+A1[1,0,0] Derivative[0,2,0][A111][1,0,0];
Derivative[1,0,1][A1][1,0,0]:=A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[1,0,0][A111][1,0,0]+A1[1,0,0] Derivative[1,0,1][A111][1,0,0];
Derivative[1,1,0][A1][1,0,0]:=A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[1,0,0][A111][1,0,0]+A1[1,0,0] Derivative[1,1,0][A111][1,0,0];
Derivative[1,2,0][A1][1,0,0]:=A1[1,0,0] ((Derivative[0,1,0][A111][1,0,0])^2) Derivative[1,0,0][A111][1,0,0]+A1[1,0,0] Derivative[0,2,0][A111][1,0,0] Derivative[1,0,0][A111][1,0,0]+2 A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[1,1,0][A111][1,0,0]+A1[1,0,0] Derivative[1,2,0][A111][1,0,0];
Derivative[2,0,0][A1][1,0,0]:=A1[1,0,0] ((Derivative[1,0,0][A111][1,0,0])^2)+A1[1,0,0] Derivative[2,0,0][A111][1,0,0];
Derivative[2,0,1][A1][1,0,0]:=A1[1,0,0] Derivative[0,0,1][A111][1,0,0] ((Derivative[1,0,0][A111][1,0,0])^2)+2 A1[1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,0,1][A111][1,0,0]+A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[2,0,0][A111][1,0,0]+A1[1,0,0] Derivative[2,0,1][A111][1,0,0];
Derivative[2,1,0][A1][1,0,0]:=A1[1,0,0] Derivative[0,1,0][A111][1,0,0] ((Derivative[1,0,0][A111][1,0,0])^2)+2 A1[1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,1,0][A111][1,0,0]+A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[2,0,0][A111][1,0,0]+A1[1,0,0] Derivative[2,1,0][A111][1,0,0];
Derivative[2,1,1][A1][1,0,0]:=A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[0,1,0][A111][1,0,0] ((Derivative[1,0,0][A111][1,0,0])^2)+A1[1,0,0] Derivative[0,1,1][A111][1,0,0] ((Derivative[1,0,0][A111][1,0,0])^2)+2 A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,0,1][A111][1,0,0]+2 A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,1,0][A111][1,0,0]+2 A1[1,0,0] Derivative[1,0,1][A111][1,0,0] Derivative[1,1,0][A111][1,0,0]+2 A1[1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,1,1][A111][1,0,0]+A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[2,0,0][A111][1,0,0]+A1[1,0,0] Derivative[0,1,1][A111][1,0,0] Derivative[2,0,0][A111][1,0,0]+A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[2,0,1][A111][1,0,0]+A1[1,0,0] Derivative[0,0,1][A111][1,0,0] Derivative[2,1,0][A111][1,0,0]+A1[1,0,0] Derivative[2,1,1][A111][1,0,0];
Derivative[2,2,0][A1][1,0,0]:=A1[1,0,0] ((Derivative[0,1,0][A111][1,0,0])^2) ((Derivative[1,0,0][A111][1,0,0])^2)+A1[1,0,0] Derivative[0,2,0][A111][1,0,0] ((Derivative[1,0,0][A111][1,0,0])^2)+4 A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,1,0][A111][1,0,0]+2 A1[1,0,0] ((Derivative[1,1,0][A111][1,0,0])^2)+2 A1[1,0,0] Derivative[1,0,0][A111][1,0,0] Derivative[1,2,0][A111][1,0,0]+A1[1,0,0] ((Derivative[0,1,0][A111][1,0,0])^2) Derivative[2,0,0][A111][1,0,0]+A1[1,0,0] Derivative[0,2,0][A111][1,0,0] Derivative[2,0,0][A111][1,0,0]+2 A1[1,0,0] Derivative[0,1,0][A111][1,0,0] Derivative[2,1,0][A111][1,0,0]+A1[1,0,0] Derivative[2,2,0][A111][1,0,0];

Derivative[0,0,1][A2][1,0,1]:=A2[1,0,1]*Derivative[0,0,1][A222][1,0,1];
Derivative[0,1,0][A2][1,0,1]:=A2[1,0,1]*Derivative[0,1,0][A222][1,0,1];
Derivative[1,0,0][A2][1,0,1]:=A2[1,0,1]*Derivative[1,0,0][A222][1,0,1];
Derivative[0,1,1][A2][1,0,1]:=A2[1,0,1] Derivative[0,0,1][A222][1,0,1] Derivative[0,1,0][A222][1,0,1]+A2[1,0,1] Derivative[0,1,1][A222][1,0,1];
Derivative[0,1,2][A2][1,0,1]:=A2[1,0,1] ((Derivative[0,0,1][A222][1,0,1])^2) Derivative[0,1,0][A222][1,0,1]+A2[1,0,1] Derivative[0,0,2][A222][1,0,1] Derivative[0,1,0][A222][1,0,1]+2 A2[1,0,1] Derivative[0,0,1][A222][1,0,1] Derivative[0,1,1][A222][1,0,1]+A2[1,0,1] Derivative[0,1,2][A222][1,0,1];
Derivative[0,0,2][A2][1,0,1]:=A2[1,0,1] ((Derivative[0,0,1][A222][1,0,1])^2)+A2[1,0,1] Derivative[0,0,2][A222][1,0,1];
Derivative[1,0,1][A2][1,0,1]:=A2[1,0,1] Derivative[0,0,1][A222][1,0,1] Derivative[1,0,0][A222][1,0,1]+A2[1,0,1] Derivative[1,0,1][A222][1,0,1];
Derivative[1,1,0][A2][1,0,1]:=A2[1,0,1] Derivative[0,1,0][A222][1,0,1] Derivative[1,0,0][A222][1,0,1]+A2[1,0,1] Derivative[1,1,0][A222][1,0,1];
Derivative[1,1,1][A2][1,0,1]:=A2[1,0,1] Derivative[0,0,1][A222][1,0,1] Derivative[0,1,0][A222][1,0,1] Derivative[1,0,0][A222][1,0,1]+A2[1,0,1] Derivative[0,1,1][A222][1,0,1] Derivative[1,0,0][A222][1,0,1]+A2[1,0,1] Derivative[0,1,0][A222][1,0,1] Derivative[1,0,1][A222][1,0,1]+A2[1,0,1] Derivative[0,0,1][A222][1,0,1] Derivative[1,1,0][A222][1,0,1]+A2[1,0,1] Derivative[1,1,1][A222][1,0,1];
Derivative[2,0,0][A2][1,0,1]:=A2[1,0,1] ((Derivative[1,0,0][A222][1,0,1])^2)+A2[1,0,1] Derivative[2,0,0][A222][1,0,1];
Derivative[2,1,0][A2][1,0,1]:=A2[1,0,1] Derivative[0,1,0][A222][1,0,1] ((Derivative[1,0,0][A222][1,0,1])^2)+2 A2[1,0,1] Derivative[1,0,0][A222][1,0,1] Derivative[1,1,0][A222][1,0,1]+A2[1,0,1] Derivative[0,1,0][A222][1,0,1] Derivative[2,0,0][A222][1,0,1]+A2[1,0,1] Derivative[2,1,0][A222][1,0,1];

Derivative[0,0,1][A3][1,1,1]:=A3[1,1,1]*Derivative[0,0,1][A333][1,1,1];
Derivative[0,1,0][A3][1,1,1]:=A3[1,1,1]*Derivative[0,1,0][A333][1,1,1];
Derivative[1,0,0][A3][1,1,1]:=A3[1,1,1]*Derivative[1,0,0][A333][1,1,1];Derivative[0,0,2][A3][1,1,1]:=A3[1,1,1] ((Derivative[0,0,1][A333][1,1,1])^2)+A3[1,1,1] Derivative[0,0,2][A333][1,1,1];
Derivative[0,1,1][A3][1,1,1]:=A3[1,1,1] Derivative[0,0,1][A333][1,1,1] Derivative[0,1,0][A333][1,1,1]+A3[1,1,1] Derivative[0,1,1][A333][1,1,1];
Derivative[0,2,0][A3][1,1,1]:=A3[1,1,1] ((Derivative[0,1,0][A333][1,1,1])^2)+A3[1,1,1] Derivative[0,2,0][A333][1,1,1];
Derivative[1,0,1][A3][1,1,1]:=A3[1,1,1] Derivative[0,0,1][A333][1,1,1] Derivative[1,0,0][A333][1,1,1]+A3[1,1,1] Derivative[1,0,1][A333][1,1,1];
Derivative[1,1,0][A3][1,1,1]:=A3[1,1,1] Derivative[0,1,0][A333][1,1,1] Derivative[1,0,0][A333][1,1,1]+A3[1,1,1] Derivative[1,1,0][A333][1,1,1];
Derivative[2,0,0][A3][1,1,1]:=A3[1,1,1] ((Derivative[1,0,0][A333][1,1,1])^2)+A3[1,1,1] Derivative[2,0,0][A333][1,1,1];

A1[1,0,0]:=0.2177787166195363783230075141194468131307977550013559376482764035236264911122526205579254438235637656918339357748032`100.;
A2[1,0,1]:=A1[1,0,0];
A3[1,1,1]:=A1[1,0,0];

Derivative[0,1,0][A111][1,0,0]:=0.84302205786030997641986237643396486410003074664813325372966418756711926688762159124159165565392`75.
Derivative[0,0,1][A111][1,0,0]:=2*Derivative[0,1,0][A111][1,0,0];
Derivative[0,1,1][A111][1,0,0]:=-1.190727816059283072434053614108579419745932058046617974308297869696307135808149094290370550743993`75.
Derivative[0,2,0][A111][1,0,0]:=-1.450209087855249540828877087187312382773836051487588631246608261693943620465558872769919442550403`75.;
Derivative[1,0,0][A111][1,0,0]:=3*Derivative[0,1,0][A111][1,0,0];
Derivative[1,0,1][A111][1,0,0]:=-4.326149391201720525313021068795706331429266638208531355130930119053118430049057281918885962161142`75.
Derivative[1,1,1][A111][1,0,0]:=4.922546395398470985919121032181312118733181980067924933193051055404874059727834172602287213077508`75.;
Derivative[1,0,2][A111][1,0,0]:=2*Derivative[1,1,1][A111][1,0,0];
Derivative[1,1,0][A111][1,0,0]:=Derivative[1,0,1][A111][1,0,0]/2;
Derivative[1,2,0][A111][1,0,0]:=5.203442844714798964018612580621705162127001410139188853357799078660730898691806653152754725469967`75.;
Derivative[2,0,0][A111][1,0,0]:=-2.793739632789949812117690423089539370154084193816941952109962433096011953452217955081858689463119`75.;
Derivative[2,1,0][A111][1,0,0]:=4.641649946082143007819629483740919075339362549996661013028303032149017220763861683754510494643289`75.;
Derivative[2,0,1][A111][1,0,0]:=2*Derivative[2,1,0][A111][1,0,0];
Derivative[0,0,2][A111][1,0,0]:=2*Derivative[0,1,1][A111][1,0,0];
Derivative[2,1,1][A111][1,0,0]:=-16.662094909253587645981626246039311879645863234835104449613586577081516802089496923754264885014504`75.
Derivative[2,2,0][A111][1,0,0]:=-16.136985712555634435795524534154684996066912955460012951066400738858144247842042449094068963096076`75.;

Derivative[0,1,0][A222][1,0,1]:=Derivative[0,1,0][A111][1,0,0];
Derivative[0,0,1][A222][1,0,1]:=2*Derivative[0,1,0][A111][1,0,0];
Derivative[1,0,0][A222][1,0,1]:=Derivative[0,1,0][A111][1,0,0];
Derivative[0,1,1][A222][1,0,1]:=Derivative[0,1,1][A111][1,0,0];
Derivative[0,0,2][A222][1,0,1]:=Derivative[0,0,2][A111][1,0,0];
Derivative[0,1,2][A222][1,0,1]:=2.253473304856102095489192873464602805750241458417489508846347471733613639206760865244491245099618`75.;
Derivative[1,0,1][A222][1,0,1]:=2*Derivative[1,1,0][A222][1,0,1];
Derivative[1,1,0][A222][1,0,1]:=-0.972346879541577190222456920289273745968701261057647703257167189830252079216379548231505572162349`75.;
Derivative[2,0,0][A222][1,0,1]:=3.477103517494925093640244486284714453212584966506884809535302065617610635029598419913002810379413`75.;
Derivative[1,1,1][A222][1,0,1]:=2.66907309054236889042992815871670931298294052165043542434670358367162303989157252614800129547396`75.;
Derivative[2,1,0][A222][1,0,1]:=-2.949969539858696868529419707157102356376759951721699344511451606927479878855545008894380831862888`75.;

Derivative[0,0,1][A333][1,1,1]:=Derivative[0,1,0][A111][1,0,0];
Derivative[0,1,0][A333][1,1,1]:=Derivative[0,0,1][A333][1,1,1];
Derivative[0,0,2][A333][1,1,1]:=Derivative[0,2,0][A111][1,0,0];
Derivative[0,2,0][A333][1,1,1]:=Derivative[0,2,0][A111][1,0,0];
Derivative[0,1,1][A333][1,1,1]:=0.259481271795966468394823473078732963027903993440970656938310391997636484657409776724613359110604`75.;
Derivative[1,1,0][A333][1,1,1]:=Derivative[1,1,0][A222][1,0,1];
Derivative[1,0,1][A333][1,1,1]:=Derivative[1,1,0][A333][1,1,1];
Derivative[1,0,0][A333][1,1,1]:=Derivative[0,1,0][A111][1,0,0];
Derivative[2,0,0][A333][1,1,1]:=Derivative[2,0,0][A222][1,0,1];

Definitions

A11[s_,w1_,w2_]:=(1-(p^(-1-w2))) (1-(p^(-1-w1-w2))) (1+(((((1-(p^(-s))))^3) ((1/(-1+(p^(1+w2))))+(1/(-1+(p^(1+w1+w2))))+(1/((-1+(p^(1+w2))) (-1+(p^(1+w1+w2)))))))/(1-(1/p))))//.p->Prime[n];
A22[s_,w1_,w2_]:=(1-(p^(s-1-w2))) (1-(p^(s-1-w1-w2))) (1+(((((1-(p^(-s))))^3) ((1/(-1+(p^(1+w2-s))))+(1/(-1+(p^(1+w1+w2-s))))+(1/((-1+(p^(1+w2-s))) (-1+(p^(1+w1+w2-s)))))))/(1-(1/p))))//.p->Prime[n];
A33[s_,w1_,w2_]:=(1-(p^(s-1-w2))) (1-(p^(s-1-w1))) (1+(((((1-(p^(-s))))^3) ((1/(-1+(p^(1+w2-s))))+(1/(-1+(p^(1+w1-s))))+(1/((-1+(p^(1+w2-s))) (-1+(p^(1+w1-s)))))))/(1-(1/p))))//.p->Prime[n];

Computations of the constants

Constants from the first residue

Derivative[0,1,0][A111][1,0,0]:=0.84302205786030997641986237643396486410003074664813325372966418756711926688762159124159165565392`75.;

Simplify[D[Log[A11[s,w1,w2]],w1]//.{s->1,w1->0,w2->0}]

N[Sum[((Log[Prime[n]] (-1+2 Prime[n]))/(1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3))),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[(( (-1+2 Prime[n]))/(1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'[k]+Log[2]/2^k),{k,1,Length[CC]}]+((Log[Prime[n]] (-1+2 Prime[n]))/(1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))//.Prime[n]->2,75]],{t,1000,1500,100}]]

Derivative[0,1,0][A111][1,0,0]:=0.84302205786030997641986237643396486410003074664813325372966418756711926688762159124159165565392`75.;

Derivative[0,0,1][A111][1,0,0]:=2*Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A11[s,w1,w2]],w2]//.{s->1,w1->0,w2->0}]

Derivative[0,0,1][A111][1,0,0]:=2*Derivative[0,1,0][A111][1,0,0];

Derivative[0,1,1][A111][1,0,0]:=-1.190727816059283072434053614108579419745932058046617974308297869696307135808149094290370550743993`75.;

Simplify[D[D[Log[A11[s,w1,w2]],w1],w2]//.{s->1,w1->0,w2->0}]

N[Sum[-((((Log[Prime[n]])^2) Prime[n] (-1+Prime[n]+2 ((Prime[n])^2)))/((-1+Prime[n]) (((-1+2 Prime[n]+((Prime[n])^2)))^2))),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[-(( Prime[n] (-1+Prime[n]+2 ((Prime[n])^2)))/((-1+Prime[n]) (((-1+2 Prime[n]+((Prime[n])^2)))^2)))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k-Log[3]^2/3^k),{k,1,Length[CC]}]-((18 ((Log[2])^2))/49)-((15 ((Log[3])^2))/98),75]],{t,1000,1200,50}]]

Derivative[0,1,1][A111][1,0,0]:=-1.190727816059283072434053614108579419745932058046617974308297869696307135808149094290370550743993`75.;

Derivative[0,2,0][A111][1,0,0]:=-1.450209087855249540828877087187312382773836051487588631246608261693943620465558872769919442550403`75.;

Simplify[D[D[Log[A11[s,w1,w2]],w1],w1]//.{s->1,w1->0,w2->0}]

-((((Log[Prime[n]])^2) Prime[n] (1-3 Prime[n]+((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))

N[Sum[-((((Log[Prime[n]])^2) Prime[n] (1-3 Prime[n]+((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2)),{n,1,10^5}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[-((Prime[n] (1-3 Prime[n]+((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]-((((Log[Prime[n]])^2) Prime[n] (1-3 Prime[n]+((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.n->1,75]],{t,950,1000,25}]]

Derivative[0,2,0][A111][1,0,0]:=-1.450209087855249540828877087187312382773836051487588631246608261693943620465558872769919442550403`75.;

Derivative[1,0,0][A111][1,0,0]:=3*Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A11[s,w1,w2]],s]//.{s->1,w1->0,w2->0}]

Derivative[1,0,0][A111][1,0,0]:=3*Derivative[0,1,0][A111][1,0,0];

Derivative[1,0,1][A111][1,0,0]:=-4.326149391201720525313021068795706331429266638208531355130930119053118430049057281918885962161142`75.;

Simplify[D[D[Log[A11[s,w1,w2]],s],w2]//.{s->1,w1->0,w2->0}]

N[Sum[-((6 ((Log[Prime[n]])^2) ((Prime[n])^4))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[-((6  ((Prime[n])^4))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]-((6 ((Log[Prime[n]])^2) ((Prime[n])^4))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->2,75]],{t,1000,1100,50}]]

Derivative[1,0,1][A111][1,0,0]:=-4.326149391201720525313021068795706331429266638208531355130930119053118430049057281918885962161142`75.;

Derivative[1,1,1][A111][1,0,0]:=4.922546395398470985919121032181312118733181980067924933193051055404874059727834172602287213077508`75.;

Simplify[D[D[D[Log[A11[s,w1,w2]],s],w1],w2]//.{s->1,w1->0,w2->0}]

N[Sum[((3 ((Log[Prime[n]])^3) ((Prime[n])^4) (-2+3 Prime[n]+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[((3 ((Prime[n])^4) (-2+3 Prime[n]+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+((3 ((Log[Prime[n]])^3) ((Prime[n])^4) (-2+3 Prime[n]+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->2,75]],{t,1000,1100,50}]]

Derivative[1,1,1][A111][1,0,0]:=4.922546395398470985919121032181312118733181980067924933193051055404874059727834172602287213077508`75.;

Derivative[1,0,2][A111][1,0,0]:=2*Derivative[1,1,1][A111][1,0,0];

Simplify[D[D[D[Log[A11[s,w1,w2]],s],w2],w2]//.{s->1,w1->0,w2->0}]

Derivative[1,0,2][A111][1,0,0]:=2*Derivative[1,1,1][A111][1,0,0];

Derivative[1,1,0][A111][1,0,0]:=Derivative[1,0,1][A111][1,0,0]/2;

Simplify[D[D[Log[A11[s,w1,w2]],s],w1]//.{s->1,w1->0,w2->0}]

Derivative[1,1,0][A111][1,0,0]:=Derivative[1,0,1][A111][1,0,0]/2;

Derivative[1,2,0][A111][1,0,0]:=5.203442844714798964018612580621705162127001410139188853357799078660730898691806653152754725469967`75.;

Simplify[D[D[D[Log[A11[s,w1,w2]],s],w1],w1]//.{s->1,w1->0,w2->0}]

N[Sum[((3 ((Log[Prime[n]])^3) ((Prime[n])^4) (-1+Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[((3  ((Prime[n])^4) (-1+Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+((3 ((Log[Prime[n]])^3) ((Prime[n])^4) (-1+Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->2,75]],{t,1000,1100,50}]]

Derivative[1,2,0][A111][1,0,0]:=5.203442844714798964018612580621705162127001410139188853357799078660730898691806653152754725469967`75.;

Derivative[2,0,0][A111][1,0,0]:=-2.793739632789949812117690423089539370154084193816941952109962433096011953452217955081858689463119`75.;

Simplify[D[D[Log[A11[s,w1,w2]],s],s]//.{s->1,w1->0,w2->0}]

N[Sum[-((3 ((Log[Prime[n]])^2) Prime[n] (-1+2 Prime[n]) (-1-Prime[n]+((Prime[n])^2)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[-((3  Prime[n] (-1+2 Prime[n]) (-1-Prime[n]+((Prime[n])^2)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]-((3 ((Log[Prime[n]])^2) Prime[n] (-1+2 Prime[n]) (-1-Prime[n]+((Prime[n])^2)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->2,75]],{t,1000,1100,50}]]

Derivative[2,0,0][A111][1,0,0]:=-2.793739632789949812117690423089539370154084193816941952109962433096011953452217955081858689463119`75.;

Derivative[2,1,0][A111][1,0,0]:=4.641649946082143007819629483740919075339362549996661013028303032149017220763861683754510494643289`75.;

Simplify[D[D[D[Log[A11[s,w1,w2]],s],s],w1]//.{s->1,w1->0,w2->0}]

N[Sum[((3 ((Log[Prime[n]])^3) ((Prime[n])^4) (-3+5 Prime[n]-((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[((3((Prime[n])^4) (-3+5 Prime[n]-((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+((3 ((Log[Prime[n]])^3) ((Prime[n])^4) (-3+5 Prime[n]-((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->2,75]],{t,1000,1100,50}]]

Derivative[2,1,0][A111][1,0,0]:=4.641649946082143007819629483740919075339362549996661013028303032149017220763861683754510494643289`75.;

Derivative[2,0,1][A111][1,0,0]:=2*Derivative[2,1,0][A111][1,0,0];

Simplify[D[D[D[Log[A11[s,w1,w2]],s],s],w2]//.{s->1,w1->0,w2->0}]

Derivative[2,0,1][A111][1,0,0]:=2*Derivative[2,1,0][A111][1,0,0];

Derivative[0,0,2][A111][1,0,0]:=2*Derivative[0,1,1][A111][1,0,0];

Simplify[D[D[Log[A11[s,w1,w2]],w2],w2]//.{s->1,w1->0,w2->0}]

Derivative[0,0,2][A111][1,0,0]:=2*Derivative[0,1,1][A111][1,0,0];

Derivative[2,1,1][A111][1,0,0]:=-16.662094909253587645981626246039311879645863234835104449613586577081516802089496923754264885014504`75.;

Simplify[D[D[D[D[Log[A11[s,w1,w2]],s],s],w1],w2]//.{s->1,w1->0,w2->0}]

N[Sum[-((3 ((Log[Prime[n]])^4) ((Prime[n])^4) (6-19 Prime[n]+17 ((Prime[n])^2)-8 ((Prime[n])^3)+20 ((Prime[n])^4)-((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^4)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1500},Do[CC=Join[{0},Series[-((3 ((Prime[n])^4) (6-19 Prime[n]+17 ((Prime[n])^2)-8 ((Prime[n])^3)+20 ((Prime[n])^4)-((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^4))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''''[k]-Log[2]^4/2^k),{k,1,Length[CC]}]+-((3 ((Log[Prime[n]])^4) ((Prime[n])^4) (6-19 Prime[n]+17 ((Prime[n])^2)-8 ((Prime[n])^3)+20 ((Prime[n])^4)-((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^4))//.Prime[n]->2,75]],{t,950,975,25}]]

Derivative[2,1,1][A111][1,0,0]:=-16.662094909253587645981626246039311879645863234835104449613586577081516802089496923754264885014504`75.;

Derivative[2,2,0][A111][1,0,0]:=-16.136985712555634435795524534154684996066912955460012951066400738858144247842042449094068963096076`75.;

Simplify[D[D[D[D[Log[A11[s,w1,w2]],s],s],w1],w1]//.{s->1,w1->0,w2->0}]

N[Sum[-((3 ((Log[Prime[n]])^4) ((Prime[n])^4) (3-8 Prime[n]+3 ((Prime[n])^2)+11 ((Prime[n])^4)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^4)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1500},Do[CC=Join[{0},Series[-((3  ((Prime[n])^4) (3-8 Prime[n]+3 ((Prime[n])^2)+11 ((Prime[n])^4)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^4))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''''[k]-Log[2]^4/2^k),{k,1,Length[CC]}]+-((3 ((Log[Prime[n]])^4) ((Prime[n])^4) (3-8 Prime[n]+3 ((Prime[n])^2)+11 ((Prime[n])^4)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^4))//.Prime[n]->2,75]],{t,950,975,25}]]

Derivative[2,2,0][A111][1,0,0]:=-16.136985712555634435795524534154684996066912955460012951066400738858144247842042449094068963096076`75.;

Computing the 1st residue

R1:=N[Simplify[3Residue[Residue[Residue[f1[s,w1,w2],{s,1}],{w2,0}],{w1,0}]],100];
R1

Constants from the 2nd residue

Derivative[0,1,0][A222][1,0,1]:=Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A22[s,w1,w2]],w1]//.{s->1,w1->0,w2->1}]

Derivative[0,1,0][A222][1,0,1]:=Derivative[0,1,0][A111][1,0,0];

Derivative[0,0,1][A222][1,0,1]:=2*Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A22[s,w1,w2]],w2]//.{s->1,w2->1,w1->0}]

Derivative[0,0,1][A222][1,0,1]:=2*Derivative[0,1,0][A111][1,0,0];

Derivative[1,0,0][A222][1,0,1]:=Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A22[s,w1,w2]],s]//.{s->1,w1->0,w2->1}]

Derivative[1,0,0][A222][1,0,1]:=Derivative[0,1,0][A111][1,0,0];

Derivative[0,1,1][A222][1,0,1]:=Derivative[0,1,1][A111][1,0,0];

Simplify[D[D[Log[A22[s,w1,w2]],w1],w2]//.{s->1,w1->0,w2->1}]

Derivative[0,1,1][A222][1,0,1]:=Derivative[0,1,1][A111][1,0,0];

Derivative[0,1,2][A222][1,0,1]:=2.253473304856102095489192873464602805750241458417489508846347471733613639206760865244491245099618`75.;

Simplify[D[D[D[Log[A22[s,w1,w2]],w1],w2],w2]//.{s->1,w1->0,w2->1}]

N[Sum[((((Log[Prime[n]])^3) Prime[n] (-1+Prime[n]+6 ((Prime[n])^2)-6 ((Prime[n])^3)-3 ((Prime[n])^4)-3 ((Prime[n])^5)+2 ((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[((Prime[n] (-1+Prime[n]+6 ((Prime[n])^2)-6 ((Prime[n])^3)-3 ((Prime[n])^4)-3 ((Prime[n])^5)+2 ((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+((((Log[Prime[n]])^3) Prime[n] (-1+Prime[n]+6 ((Prime[n])^2)-6 ((Prime[n])^3)-3 ((Prime[n])^4)-3 ((Prime[n])^5)+2 ((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->2,75]],{t,750,750,50}]]

Derivative[0,1,2][A222][1,0,1]:=2.253473304856102095489192873464602805750241458417489508846347471733613639206760865244491245099618`75.;

Derivative[0,0,2][A222][1,0,1]:=Derivative[0,0,2][A111][1,0,0];

Simplify[D[D[Log[A22[s,w1,w2]],w2],w2]//.{s->1,w1->0,w2->1}]

Derivative[0,0,2][A222][1,0,1]:=Derivative[0,0,2][A111][1,0,0];

Derivative[1,0,1][A222][1,0,1]:=2*Derivative[1,1,0][A222][1,0,1];

Simplify[D[D[Log[A22[s,w1,w2]],s],w2]//.{s->1,w1->0,w2->1}]

Derivative[1,0,1][A222][1,0,1]:=2*Derivative[1,1,0][A222][1,0,1];

Derivative[1,1,0][A222][1,0,1]:=-0.972346879541577190222456920289273745968701261057647703257167189830252079216379548231505572162349`75.;

Simplify[D[D[Log[A22[s,w1,w2]],s],w1]//.{s->1,w1->0,w2->1}]

-((((Log[Prime[n]])^2) Prime[n] (-1+2 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))
-((((Log[Prime[n]])^2) Prime[n] (-1+2 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->2

N[Sum[-((((Log[Prime[n]])^2) Prime[n] (-1+2 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[-((Prime[n] (-1+2 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]-(30/49) ((Log[2])^2),75]],{t,900,950,25}]]

Derivative[1,1,0][A222][1,0,1]:=-0.972346879541577190222456920289273745968701261057647703257167189830252079216379548231505572162349`75.;

Derivative[1,1,1][A222][1,0,1]:=2.66907309054236889042992815871670931298294052165043542434670358367162303989157252614800129547396`75.;

Simplify[D[D[D[Log[A22[s,w1,w2]],s],w1],w2]//.{s->1,w1->0,w2->1}]

N[Sum[((((Log[Prime[n]])^3) Prime[n] (1-Prime[n]-6 ((Prime[n])^2)+12 ((Prime[n])^4)+3 ((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[((Prime[n] (1-Prime[n]-6 ((Prime[n])^2)+12 ((Prime[n])^4)+3 ((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+((((Log[Prime[n]])^3) Prime[n] (1-Prime[n]-6 ((Prime[n])^2)+12 ((Prime[n])^4)+3 ((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->2,75]],{t,900,1000,50}]]

Derivative[1,1,1][A222][1,0,1]:=2.66907309054236889042992815871670931298294052165043542434670358367162303989157252614800129547396`75.;

Derivative[2,0,0][A222][1,0,1]:=3.477103517494925093640244486284714453212584966506884809535302065617610635029598419913002810379413`75.;

Simplify[D[D[Log[A22[s,w1,w2]],s],s]//.{s->1,w1->0,w2->1}]

N[Sum[((((Log[Prime[n]])^2) Prime[n] (-5+7 Prime[n]+11 ((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1250},Do[CC=Join[{0},Series[((Prime[n] (-5+7 Prime[n]+11 ((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]+((((Log[Prime[n]])^2) Prime[n] (-5+7 Prime[n]+11 ((Prime[n])^2)+2 ((Prime[n])^3)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->2,75]],{t,900,1000,50}]]

Derivative[2,0,0][A222][1,0,1]:=3.477103517494925093640244486284714453212584966506884809535302065617610635029598419913002810379413`75.;

Derivative[2,1,0][A222][1,0,1]:=-2.949969539858696868529419707157102356376759951721699344511451606927479878855545008894380831862888`75.;

Simplify[D[D[D[Log[A22[s,w1,w2]],s],s],w1]//.{s->1,w1->0,w2->1}]

N[Sum[-((((Log[Prime[n]])^3) Prime[n] (1-Prime[n]-6 ((Prime[n])^2)+3 ((Prime[n])^3)+6 ((Prime[n])^4)+6 ((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3)),{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1500},Do[CC=Join[{0},Series[-((Prime[n] (1-Prime[n]-6 ((Prime[n])^2)+3 ((Prime[n])^3)+6 ((Prime[n])^4)+6 ((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+-((((Log[Prime[n]])^3) Prime[n] (1-Prime[n]-6 ((Prime[n])^2)+3 ((Prime[n])^3)+6 ((Prime[n])^4)+6 ((Prime[n])^5)+((Prime[n])^6)))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^3))//.Prime[n]->2,75]],{t,950,975,25}]]

Derivative[2,1,0][A222][1,0,1]:=-2.949969539858696868529419707157102356376759951721699344511451606927479878855545008894380831862888`75.;

Computing the 2nd residue

Simplify[-3Residue[Residue[Residue[f2[s,w1,w2],{s,1}],{w2,1}],{w1,0}]]

R2:=N[Simplify[-3Residue[Residue[Residue[f2[s,w1,w2],{s,1}],{w2,1}],{w1,0}]],75];
R2

Constants from the last residue

Simplify[A3[1,1,1]]

Derivative[0,0,1][A333][1,1,1]:=Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A33[s,w1,w2]],w2]//.{s->1,w1->1,w2->1}]

Derivative[0,0,1][A333][1,1,1]:=Derivative[0,1,0][A111][1,0,0];

Derivative[0,1,0][A333][1,1,1]:=Derivative[0,0,1][A333][1,1,1];

Simplify[D[Log[A33[s,w1,w2]],w1]//.{s->1,w1->1,w2->1}]

Derivative[0,1,0][A333][1,1,1]:=Derivative[0,0,1][A333][1,1,1];

Derivative[0,0,2][A333][1,1,1]:=Derivative[0,2,0][A111][1,0,0];

Simplify[D[D[Log[A33[s,w1,w2]],w2],w2]//.{s->1,w1->1,w2->1}]

Derivative[0,0,2][A333][1,1,1]:=Derivative[0,2,0][A111][1,0,0];

Derivative[0,2,0][A333][1,1,1]:=Derivative[0,2,0][A111][1,0,0];

Simplify[D[D[Log[A33[s,w1,w2]],w1],w1]//.{s->1,w1->1,w2->1}]

Derivative[0,2,0][A333][1,1,1]:=Derivative[0,2,0][A111][1,0,0];

Derivative[0,1,1][A333][1,1,1]:=0.259481271795966468394823473078732963027903993440970656938310391997636484657409776724613359110604`75.;

Simplify[D[D[Log[A33[s,w1,w2]],w1],w2]//.{s->1,w1->1,w2->1}]

N[Sum[((((Log[Prime[n]])^2) ((Prime[n])^2) (-1+2 Prime[n]))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2)),{n,1,10^5}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0,0},Series[(( ((Prime[n])^2) (-1+2 Prime[n]))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.Prime[n]->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]+((((Log[Prime[n]])^2) ((Prime[n])^2) (-1+2 Prime[n]))/(((1-3 Prime[n]+((Prime[n])^2)+((Prime[n])^3)))^2))//.n->1,75]],{t,1000,1100,50}]]

Derivative[0,1,1][A333][1,1,1]:=0.259481271795966468394823473078732963027903993440970656938310391997636484657409776724613359110604`75.;

Derivative[1,1,0][A333][1,1,1]:=Derivative[1,1,0][A222][1,0,1];

Simplify[D[D[Log[A33[s,w1,w2]],w1],s]//.{s->1,w1->1,w2->1}]

Derivative[1,1,0][A333][1,1,1]:=Derivative[1,1,0][A222][1,0,1];

Derivative[1,0,1][A333][1,1,1]:=Derivative[1,1,0][A333][1,1,1];

Simplify[D[D[Log[A33[s,w1,w2]],w2],s]//.{s->1,w1->1,w2->1}]

Derivative[1,0,1][A333][1,1,1]:=Derivative[1,1,0][A333][1,1,1];

Derivative[1,0,0][A333][1,1,1]:=Derivative[0,1,0][A111][1,0,0];

Simplify[D[Log[A33[s,w1,w2]],s]//.{s->1,w1->1,w2->1}]

Derivative[1,0,0][A333][1,1,1]:=Derivative[0,1,0][A111][1,0,0];

Derivative[2,0,0][A333][1,1,1]:=Derivative[2,0,0][A222][1,0,1];

Simplify[D[D[Log[A33[s,w1,w2]],s],s]//.{s->1,w1->1,w2->1}]

Derivative[2,0,0][A333][1,1,1]:=Derivative[2,0,0][A222][1,0,1];

Computing the 3rd residue

R3:=N[Simplify[Residue[Residue[Residue[f3[s,w1,w2],{s,1}],{w2,1}],{w1,1}]],75];
R3