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 search more complete list of degree 3 L-functions, or browse by underlying object:

Maass form for $\GL(3)$       Symmetric square of L-function of Elliptic curve       Artin representation of dimension 3

First complex
critical zero
underlying object $N$ $\chi$ arithmetic self-dual $\delta,\nu$ $\mu$ $\delta_1,\delta_2,\delta_3$ $\mu_1,\mu_2$ $\epsilon$
6.42223Maass cusp form1-0,0,016.40312, 0.171121
5.11568number field,001
3.89928symmetric square of therank 0 elliptic curve 11.a1121-0,101
3.19363Artin representation229$\left( \frac{229}{\cdot}\right)$0,1,10,01
2.65766Maass cusp form4-0,0,08.23979, 2.641221
2.53210Maass cusp form4-0,0,09.63244, 1.37406$e(0.33333)$
2.15869symmetric square of therank 1 elliptic curve 37.a11369-0,101
1.36848tensor productof anArtin representationand aDirichlet character28625$\chi_{1145}(228,\cdot)$0,0,10,0$e(0.76431)$
1.07683symmetric square of therank 2 elliptic curve 389.a1151321-0,101
0.93372Maass cusp form4-0,0,08.95466, 2.93659$e(0.66667)$