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# Riemann $\Xi$ function

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

(where $\Gamma(s)$ 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

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

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