Definition:Completed Riemann Zeta Function

Definition
The completed Riemann zeta function is defined on the complex plane $\C$ as:


 * $\displaystyle \forall s \in \C: \xi \left({s}\right) := \begin{cases}

\frac 1 2 s \left({s - 1}\right) \pi^{-s/2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) & : \Re \left({s}\right) > 0 \\ \xi \left({1 - s}\right) & : \Re \left({s}\right) \le 0 \end{cases}$ where $\zeta \left({s}\right)$ is the Riemann zeta function.

Also known as
The completed Riemann zeta function is also known as the Riemann xi function.