L-functions can be organized by degree. All known degree 3 L-functions have a functional equation of one of the two forms \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb R}(s + \delta + i \mu) \, \Gamma_{\mathbb C}\left(s + \nu - i \frac{\mu}{2}\right) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s) \] or \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb R}(s+\delta_1 + i \mu_1) \, \Gamma_{\mathbb R}(s+\delta_2 + i \mu_2) \, \Gamma_{\mathbb R}(s+\delta_3 - i (\mu_1 + \mu_2)) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] and an Euler product of the form \[ L(s)= \prod_{p|N} \left(1-a_p\, p^{-s} + (a_p^2 - a_{p^2})\, p^{-2s}\right)^{-1}. \prod_{p\nmid N} \left(1-a_p\, p^{-s}+\chi(p) \, \overline{a_p} \, p^{-2s} -\chi(p) \, p^{-3s}\right)^{-1}. \] Here $N$ is the conductor of the L-function, and $\chi$, known as the central character, is a primitive Dirichlet character of conductor dividing $N$. The parameters $\nu$ and $\delta$ are integers or half-integers, the parameters $\mu_j$ and $\mu$ are real, and $\delta_j$ is 0 or 1. If the central character is even (or odd) then either $2\nu + 1 + \delta$ or $\delta_1 + \delta_2 + \delta_3$ is even (or odd).

Browse Degree 3 L-functions

A sample of degree 3 L-functions is in the table below. You can also browse by underlying object:

Maass form for $\GL(3)$     Symmetric square of of an elliptic curve     Artin representation of dimension 3

First complex
critical zero
underlying object $N$ $\chi$ arithmetic self-dual $\delta,\nu$ $\mu$ $\delta_1,\delta_2,\delta_3$ $\mu_1,\mu_2$ $\varepsilon$
6.42223$\SL(3,\Z)$ Maass form110,0,016.40312, 0.171121
5.11568global number field23$\left(\frac{-23}{\cdot}\right)$$0$,001
3.89928$\Sym^2$ of an elliptic curve1211$0$,101
3.19363Artin representation229$\left(\frac{229}{\cdot}\right)$0,1,10, 01
2.65766$\GL(3)$ Maass form410,0,08.23979, 2.641221
2.53210$\GL(3)$ Maass form410,0,09.63244, 1.37406$e(1/3)$
2.15869$\Sym^2$ of an elliptic curve13691$0$,101
1.07683$\Sym^2$ of an elliptic curve1513211$0$,101
0.93372$\GL(3)$ Maass form410,0,08.95466, 2.93659$e(2/3)$