# $\xi$ and $Z$ functions

The functional equation can be written in a form which is more symmetric:

Here is known as the Riemann -function. It is an entire function of order 1, and all of its zeros lie in the critical strip.

The -function associated to a general -function is similar, except that the factor is omitted, since its only purpose was to cancel the pole at .

The function just involves a change of variables: . The functional equation now asserts that .

The Hardy -function is defined as follows. Let

and define

Then is real for real , and .

Plots of are a nice way to picture the -function on the critical line.  is called RiemannSiegelZ[t] in Mathematica.

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