Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$$ There are probably countless proofs of Riemann's functional equation, but as of yet there's not a single place where they're all concentrated.

Anyone care to share some particularly pretty proofs?

$\zeta(s)$: Riemann zeta function.

$\Gamma(s)$: Gamma function.

AFAIK none of proofs is very short and easy. I'll post just a rough sketch (in particular all analytical issues are silently ignored) of a Riemann's original proof based on the Poisson summation formula.

Let's define $$ \xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ Riemann's functional equation takes the form $$ \xi(s)=\xi(1-s). $$

By definition $$ \xi(s)= \sum_{n=1}^\infty\pi^{-s/2}n^{-s}\int_0^\infty e^{-t}t^{s/2}\frac{dt}t $$ and after substitution $t=\pi n^2x$ we get $$ \xi(s)= \int_0^\infty x^{s/2}\sum_{n=1}^\infty e^{-\pi n^2x}\frac{dx}x= \frac12\int_0^\infty x^{s/2}(\theta(x)-1)\frac{dx}x, $$ where $\theta(x)=\sum_{n\in\mathbb Z}e^{-\pi n^2x}$.

By Poisson summation formula this theta function satisfies $$ \theta(x^{-1})=x^{1/2}\theta(x), $$ which allows us to rewrite last integral in the form \begin{multline} \xi(s)= \frac12\int_1^\infty x^{s/2}(\theta(x)-1)\frac{dx}x+ \frac12\int_1^\infty x^{-s/2}(x^{1/2}\theta(x)-1)\frac{dx}x=\\ =\frac12\int_1^\infty(x^{s/2}+x^{(1-s)/2})(\theta(x)-1)\frac{dx}x-\frac1{s(1-s)}, \end{multline} which is manifestly symmetric under change $s\mapsto1-s$.

Summary:

1. The functional equation takes simpler form for the $\xi$-function.
2. The $\xi$-function is (more or less) the Mellin transform of $\theta$-function.
3. The $\theta$-function satisfies some modularity functional equation coming from the Poisson summation formula.

...or to make a long story short,

Riemann's functional equation is the Mellin transform of the Poisson summation formula.