\begin{definition}[Definition:Completed Riemann Zeta Function]
The '''completed Riemann zeta function''' $\xi: \C \to \C$ is defined on the complex plane $\C$ as:
:$\ds \forall s \in \C: \map \xi x := \begin{cases}
\dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s & : \map \Re s > 0 \\
\map \xi {1 - s} & : \map \Re s \le 0
\end{cases}$
where $\map \zeta s$ is the Riemann zeta function.
\end{definition}