Riemann xi function

The English used in this article or section may not be easy for everybody to understand. You can help Wikipedia by reading Wikipedia:How to write Simple English pages, then simplifying the article. (May 2026)

The Riemann xi function, sometimes called the completed zeta function, is an extension of the Riemann zeta function defined as ${\displaystyle \xi (s)={\frac {1}{2}}s(s-1)\pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)}$ for ${\displaystyle s\in \mathbb {C} }$ (s is a complex variable). It is defined to satisfy the functional equation

${\displaystyle \xi (1-s)=\xi (s).}$

This shows that the Riemann xi function is symmetric about the critical line Re(s)= 1/2. Unlike the Riemann zeta function which is a meromorphic function on the complex plane with a simple pole (undefined value that can be cancelled by multiplying by (s-1)) at s=1, the Riemann xi function is an entire function. This means that it is holomorphic (infinitely complex differentiable and analytic/equal to its own Taylor series) everywhere on the complex plane.

References

This short article about mathematics can be made longer. You can help Wikipedia by adding to it.