If we put the function $ f(x)= |x|^{s-1} $ inside the Poisson sum formula and consider that
$ \sum_{n=1}n^{z-1}= \zeta (1-s) $ then we can easily give a proof of Riemann's functional equation
$$ \zeta(1-s)= 2(2\pi )^{-s}\Gamma (s)\cos(\pi s/2)\zeta (s) $$
but why this method work ? We have used the zeta regularization of the series but the series is not convergent anyway the method seems to work.