Riemann $\mathrm{\Xi }$ function

The Riemann Xi function

$$\mathrm{\Xi }(s)={\pi }^{-\frac{1}{2}s}\mathrm{\Gamma }(\frac{1}{2}s)\zeta (s),$$

(where $\mathrm{\Gamma }(s)$ is Euler’s Gamma function and $\zeta (s)$ is the Riemann zeta function), is the key to the functional equation for the Riemann zeta function.

Riemann himself used the notation of a lower case xi ($\xi$). The famous Riemann hypothesis is equivalent to the assertion that all the zeros of $\xi$ are real, in fact Riemann himself presented his original hypothesis in terms of that function.

Riemann’s lower case xi is defined as

$$\xi (s)=\frac{1}{2}s(s-1)\mathrm{\Xi }(s).$$