# functional equation of the Riemann zeta function

Let $\mathrm{\Gamma}$ denote the gamma function^{}, $\zeta $ the Riemann zeta function^{} and $s$ any complex number^{}. Then

${\pi}^{\frac{s-1}{2}}\mathrm{\Gamma}\left({\displaystyle \frac{1-s}{2}}\right)\zeta (1-s)={\pi}^{\frac{-s}{2}}\mathrm{\Gamma}\left({\displaystyle \frac{s}{2}}\right)\zeta (s).$ |

Though the equation appears too intricate to be of any use, the inherent in the formula makes this the simplest method of evaluating $\zeta (s)$ at points to the left of the critical strip^{}.

Title | functional equation of the Riemann zeta function |
---|---|

Canonical name | FunctionalEquationOfTheRiemannZetaFunction |

Date of creation | 2013-03-22 13:54:29 |

Last modified on | 2013-03-22 13:54:29 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11M99 |

Related topic | ClassNumberFormula |