Functional Equation for Riemann Zeta Function

Theorem
Let $\zeta$ be the Riemann zeta function.

Let $\map \zeta s$ have an analytic continuation for $\map \Re s > 0$.

Then:
 * $\displaystyle \map \Gamma {\frac s 2} \pi^{-s/2} \map \zeta s = \map \Gamma {\frac {1 - s} 2} \pi^{\frac {s - 1} 2} \map \zeta {1 - s}$

where $\Gamma$ is the gamma function

Proof
Using Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function:

The functional equation:


 * $\map \xi s = \map \xi {1 - s}$

follows upon observing that this integral is invariant under $s \mapsto 1 - s$.

Also see

 * Functional Equation for Completed Riemann Zeta Function