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.
 underlying object $N$ $\chi$ arithmetic self-dual $\varepsilon$ First complexcritical zero $\Gamma_{\R}$ parameters $\Gamma_{\C}$ parameters 16.18901 $\SL(4,\Z)$ Maass form 1 1 $16.89i$, $2.272i$, $-6.035i$, $-13.13i$ 1 14.49606 $\Sp(4,\Z)$ Maass form 1 1 ✔ $4.720i$, $-4.720i$, $12.46i$, $-12.46i$ 1 4.30352 genus 2 curve,paramodular form 277 1 ✔ ✔ $\frac12$, $\frac12$ 1 3.67899 Hilbert cusp form over $\Q(\sqrt{5})$,Elliptic curve over $\Q(\sqrt5)$ 775 1 ✔ ✔ $\frac12$, $\frac12$ 1 3.50464 Artin representation 1609 $\left(\frac{1609}{\cdot}\right)$ ✔ ✔ 0, 0, 1, 1 1 2.78027 Artin representation 3215 $\left(\frac{-3215}{\cdot}\right)$ ✔ ✔ 0, 1, 1, 1 1 2.48044 Sp(4,$\mathbb Z$) Maass form 1 1 ✔ $19.81i$, $-19.81i$, $8.531i$, $-8.531i$ 1 2.26074 CM genus 2 curve,CM Hilbert cusp form over $\Q(\sqrt{5})$ 3125 1 ✔ ✔ $\frac12$, $\frac12$ 1 2.32002 $\Sym^3$ of an elliptic curve 3215 1 ✔ ✔ $\frac32$, $\frac12$ 1 1.42931 RM genus 2 curve,Hilbert cusp form over $\Q(\sqrt{5})$ 12500 1 ✔ ✔ $\frac12$, $\frac12$ 1 1.21844 QM genus 2 curve,Bianchi cusp form over $\Q(\sqrt{-3})$ 20736 1 ✔ ✔ $\frac12$, $\frac12$ 1