Functional Equation for Riemann Zeta Function

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

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

Then:
 * $\displaystyle \Gamma \left({\frac s 2}\right) \pi^{-s/2} \zeta \left({s}\right) = \Gamma \left({\frac{1-s} 2}\right) \pi^{\frac{s-1} 2} \zeta \left({1-s}\right)$

where $\Gamma$ is the gamma function

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

The functional equation:


 * $\xi \left({s}\right) = \xi \left({1 - s}\right)$

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

Also see

 * Functional Equation for Completed Riemann Zeta Function