Unsymmetric Functional Equation for Riemann Zeta Function

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

Let $\Gamma$ be the gamma function.

Then for all $s \in \C$,


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

Proof
We have for $s \notin \Z$ Euler's Reflection Formula:


 * $\displaystyle \Gamma \left({s}\right) \Gamma \left({1 - s}\right) = \frac \pi {\sin \left({\pi s}\right)}$

Replacing $s \mapsto \left({1 + s}\right) / 2$ we deduce:

Also, we have Legendre's Duplication Formula for $z \notin -\dfrac 1 2 \N_0$:


 * $\Gamma \left({s}\right) \Gamma \left({s + \dfrac 1 2}\right) = 2^{1 - 2 s} \sqrt \pi \Gamma \left({2 s}\right)$

Replacing $s \mapsto s / 2$ this yields:


 * $\Gamma \left({\dfrac s 2}\right) \Gamma \left ({\dfrac {1 + s } 2}\right) = 2^{1 - s} \sqrt \pi \Gamma \left({s}\right)$

Together these give:


 * $(1): \quad \dfrac {\Gamma \left({s / 2}\right)} {\Gamma \left({\left({1 - s}\right) / 2}\right)} = 2^{1 - s} \pi^{-1/2} \Gamma \left({s}\right) \cos \left({\pi s / 2}\right)$

Now we take the Functional Equation for Riemann Zeta Function:


 * $\pi^{-s/2} \zeta \left({s}\right) \Gamma \left({s / 2}\right) \Gamma \left({\dfrac {1 - s} 2}\right)^{-1} = \pi^{\left({s - 1}\right) / 2} \zeta \left({1 - s}\right)$

and substitute $(1)$ to give:


 * $\pi^{\left({s - 1}\right) / 2} \zeta \left({1 - s}\right) = \pi^{-\left({s + 1}\right) / 2} \zeta \left({s}\right) 2^{1 - s} \Gamma \left({s}\right) \cos \left({\pi s / 2}\right)$

Multiplying by $\pi^{(s-1)/2}$ this becomes:


 * $\zeta \left({1 - s}\right) = \pi^{-s} 2^{1 - s} \cos \left({\pi s / 2}\right) \Gamma \left({s}\right) \zeta \left({s}\right)$

as desired.