The Riemann zeta function is the prototypical L-function. It is the only L-function of degree 1 and conductor 1, and (conjecturally) it is the only primitive L-function with a pole.

## Dirichlet series

$\zeta(s)$ = 1^{} | + 2^{-s} + 3^{-s} + 4^{-s} + 5^{-s} + 6^{-s} + 7^{-s} + 8^{-s} + 9^{-s} + 10^{-s} + 11^{-s} + 12^{-s} + 13^{-s} + 14^{-s} + 15^{-s} + 16^{-s} + 17^{-s} + 18^{-s} + 19^{-s} + 20^{-s} + 21^{-s} + 22^{-s} + 23^{-s} + 24^{-s} + 25^{-s} + 26^{-s} + 27^{-s} + 28^{-s} + ⋯ |

## Functional equation

\[\begin{aligned}\xi(s)=\mathstrut &\Gamma_{\R}(s) \, \zeta(s)\cr=\mathstrut & \,\xi(1-s)\end{aligned}\]

## Invariants

Degree: | \(1\) |

Conductor: | \(1\) |

Sign: | $1$ |

Primitive: | yes |

Self-dual: | yes |

Selberg data: | \((1,\ 1,\ (0:\ ),\ 1)\) |

## Particular Values

\[\zeta(1/2) \approx -1.460354508\]

Pole at \(s=1\)

## Euler product

\(\zeta(s) = \displaystyle \prod_p (1 - p^{-s})^{-1}\)

## Imaginary part of the first few zeros on the critical line