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<h1 align=center> $\xi$ and $Z$ functions
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The \begin{rawhtml}<a href="../69a">functional equation</a>\end{rawhtml} can be written in a form which 
is more symmetric:
$$
\xi(s):=\frac12 s(s-1) \pi^{\frac12 s} \Gamma(s/2) \zeta(s)=\xi(1-s) .
$$
Here $\xi(s)$ is known as the Riemann $\xi$-function.  It is an
entire function of order 1, and all of its zeros lie in the
critical strip.

The $\xi$-function associated to a general $L$-function is similar,
except that the factor $\frac12 s(s-1)$ is omitted, since its only purpose
was to cancel the pole at $s=1$.

The $\Xi$ function just involves a change of variables:
$\Xi(z)=\xi(\frac12 + iz)$.  The functional equation now asserts
that $\Xi(z)=\Xi(-z)$.

The Hardy $Z$-function is defined as follows.  Let
$$
\vartheta=\vartheta(t)=\frac12 \arg(\chi(\frac12 + it)),
$$
and define
$$
Z(t)=e^{i\vartheta} \zeta(\frac12 + it) = 
\chi(\frac12 + it)^{-\frac12} \zeta(\frac12+it) .
$$
Then $Z(t)$ is real for real $t$, and 
$|Z(t)|=|\zeta(\frac12 + it)$.

Plots of $Z(t)$ are a nice way to picture the $\zeta$-function on
the critical line.
$Z(t)$~is called {\tt RiemannSiegelZ[t]} in Mathematica.
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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