The Riemann $\zeta$-function has functional equation

(1) |

Dirichlet $L$-functions satisfy the functional equation

(2) |

where if is even and if is odd, and .

The Dedekind zeta function of a number field
satisfies the functional equation

(3) |

Here and are the number of real and complex conjugate embeddings , is the discriminant, and is the degree of .

$L$-functions associated with a newform
satisfy the functional equation

(4) |

where if is even and if is odd.

$L$-functions associated with a Maass newform
with eigenvalue
on
satisfy the functional equation

(5) |

where if is even and if is odd.

$GL(r)$ $L$-functions satisfy functional equations
of the form

[This section needs a bit of work]

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