# Definition:Completed Riemann Zeta Function

## Definition

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.

## Also known as

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

## Also see

Thus the theory of entire functions can be applied to $\xi$.

• The factor of $\dfrac 1 2$ is convenient, though omitted in some sources.

By the Gamma Difference Equation it allows us to write:

$\map \xi s = \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2 + 1} \map \zeta s$

• None of the factors of $\xi$ except $\zeta$ have a zero in $\C \setminus \set {0, 1}$, so no information is lost about the nontrivial zeros.