Riemann xi function

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.^[1] 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.