L-functions of signature (0,0,0,0;) with real coefficients

These L-functions satisfy a functional equation with \(\Gamma\)-factors \[\begin{aligned}\Gamma_\R(s + i \mu_1)\Gamma_\R(s + i \mu_2)\Gamma_\R(s - i \mu_1)\Gamma_\R(s - i \mu_2)\end{aligned}\] with \(\mu_j\) real. By renaming and rearranging, we may assume \(0 \le \mu_2 \le \mu_1\).

L-functions of conductor 1

The dots in the plot correspond to L-functions with \((\mu_1,\mu_2)\) in the \(\Gamma\)-factors, colored according to the sign of the functional equation (blue indicates \(\epsilon=1\)). Click on any of the dots for detailed information about the L-function.


L-functions of signature (0,0,0,0;)

These L-functions satisfy a functional equation with \(\Gamma\)-factors \[\begin{aligned}\Gamma_\R(s + i \mu_1)\Gamma_\R(s + i \mu_2)\Gamma_\R(s + i \mu_3)\Gamma_\R(s + i \mu_4)\end{aligned}\] with \(\mu_j\in \R\) and \(\mu_1 + \mu_2 + \mu_3 + \mu_4 = 0\). By permuting and possibly conjugating, we may assume \(0\le \mu_2 \le \mu_1 \).

L-functions of conductor 1

The dots in the plot correspond to L-functions with \((\mu_1,\mu_2)\) in the \(\Gamma\)-factors, colored according to the sign of the functional equation (blue indicates \(\epsilon=1\)). Click on any of the dots for detailed information about the L-function.