Mellin representation of the completed Riemann zeta function

For $\Theta (u)={\sum }_{n\in \mathbb{Z}}{e}^{-\pi n^{2}u}$ and initially $\Re s>1$,The Poisson summation formula gives $\Theta (u)=u^{-1/2}\Theta (1/u)$. Splitting the integral at one therefore continues it meromorphically and makes its invariance under $s\mapsto 1-s$ visible.