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

where $\Gamma$ is the gamma function

Proof
From Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function we have:

We observe that this integral is invariant under $s \mapsto 1 - s$.

as required.

Also see

 * Functional Equation for Completed Riemann Zeta Function


 * Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function