L-functions can be organized by degree. All known degree 4 L-functions have a functional equation of one of the three forms: \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb C}(s + \nu_1) \, \Gamma_{\mathbb C}(s + \nu_2) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb R}(s+\mu_1) \, \Gamma_{\mathbb R}(s+\mu_2) \, \Gamma_{\mathbb C}\left(s+ \nu\right) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] or \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb R}(s+\mu_1) \, \Gamma_{\mathbb R}(s+\mu_2) \, \Gamma_{\mathbb R}(s+\mu_3) \, \Gamma_{\mathbb R}(s+\mu_4) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] and an Euler product of the form \[ L(s)= \prod_{p|N} \left(1-a_p\, p^{-s} + (a_p^2 - a_{p^2})\, p^{-2s} - (a_p^3 - 2 \, a_{p^2} \, a_p + a_{p^3} ) \, p^{-3s}\right)^{-1}. \prod_{p\nmid N} \left(1-a_p\, p^{-s} + (a_p^2 - a_{p^2})\, p^{-2s} + \chi(p) \, \overline{a_p} \, p^{-3s} +\chi(p) \, p^{-4s}\right)^{-1}. \] Here $N$ is the conductor of the L-function, and $\chi$, known as the central character, is a primitive Dirichlet character of conductor dividing $N$. Moreover, $\operatorname{Re}(\mu_j) \in \{0,1\}$, $\operatorname{Re}(\nu_j)$ and $\operatorname{Re}(\nu)$ are integer or an half-integer, and $2 \sum_j \mu_j + \sum_k \nu_k$ is a positive real.

Browse Degree 4 L-functions

A sample of degree 4 L-functions is in the table below. You can also search more complete list of degree 4 L-functions, or browse by underlying object:

Maass form for $\GL(4)$       Symmetric cube of L-function of Elliptic curve       Artin representation of dimension 4       Genus 2 curve over $\Q$

First complex
critical zero
$N$ $\chi$ arithmetic self-dual $\Gamma_{\R}$ $\Gamma_{\C}$ $\varepsilon$
2.32002Symmetric cube of rank 0 elliptic curve1331-$\frac{1}{2}$, $\frac{3}{2}$1
5.06823genus 2 curve169-$\frac{1}{2}$, $\frac{1}{2}$1
16.18901Maass for GL(4)1-16.89i, 2.272i, -6.035i, -13.13i1
3.67899Hilbert modular form over $\mathbb Q(\sqrt{5})$775-$\frac{1}{2}$, $\frac{1}{2}$1
14.49606Maass1-4.720i, -4.720i, 12.46i, -12.46i1
3.50464Artin1609-0, 0, 1, 11
2.48044Sp(4,$\mathbb Z$) Maass1-19.81i, -19.81i, 8.531i, -8.531i1