Riemann Xi function

In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann.

Riemann's original lower-case xi-function, ξ, has been renamed with an upper-case Xi, Ξ, by Edmund Landau (see below). Landau's lower-case xi, ξ, is defined as:

for ${\displaystyle s\in {\mathbb {C}}}$. Here ζ(s) denotes the Riemann zeta function and Γ(s) is the Gamma function. The functional equation (or reflection formula) for xi is

The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as

and obeys the functional equation

As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ. Both functions are entire and purely real for real arguments.

The general form for even integers is

where B_n denotes the n-th Bernoulli number. For example:

The ${\displaystyle \xi }$ function has the series expansion

...
Wikipedia