$\xi$ and $Z$ functions

The functional equation 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 $\vert Z(t)\vert=\vert\zeta(\frac12 + it)$.

Plots of $Z(t)$ are a nice way to picture the $\zeta$-function on the critical line. $Z(t)$ is called RiemannSiegelZ[t] in Mathematica.

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