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:

 First complexcritical zero Underlyingobject $N$ $\chi$ arithmetic self-dual $\Gamma_{\R}$ $\Gamma_{\C}$ $\varepsilon$ 2.32002 Symmetric cube of rank 0 elliptic curve 1331 - ● ● $\frac{1}{2}$, $\frac{3}{2}$ 1 5.06823 genus 2 curve 169 - ● ● $\frac{1}{2}$, $\frac{1}{2}$ 1 16.18901 Maass for GL(4) 1 - ○ ○ 16.89i, 2.272i, -6.035i, -13.13i 1 3.67899 Hilbert modular form over $\mathbb Q(\sqrt{5})$ 775 - ● ● $\frac{1}{2}$, $\frac{1}{2}$ 1 14.49606 Maass 1 - ○ ● 4.720i, -4.720i, 12.46i, -12.46i 1 3.50464 Artin 1609 - ● ● 0, 0, 1, 1 1 2.48044 Sp(4,$\mathbb Z$) Maass 1 - ○ ● 19.81i, -19.81i, 8.531i, -8.531i 1