Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.353·2-s + 0.125·4-s − 0.268·5-s − 0.431·7-s + 0.0441·8-s − 9-s − 0.0948·10-s + 0.533·13-s − 0.152·14-s + 0.0156·16-s − 1.07·17-s − 0.353·18-s + 1.06·19-s − 0.0335·20-s + 0.232·25-s + 0.188·26-s − 0.0539·28-s − 1.32·29-s + 1.06·31-s + 0.00552·32-s − 0.378·34-s + 0.115·35-s − 0.125·36-s + 0.324·37-s + 0.375·38-s − 0.0118·40-s + 1.05·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(648\) = \(2^{3} \cdot 3^{4}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 648,\ (\ :1.5, 0.5),\ 1)\) |
Particular Values
\[L(1/2, E, \mathrm{sym}^{3}) \approx 0.8470172594\]
\[L(1, E, \mathrm{sym}^{3}) \approx 1.011529058\]
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+27\ 3^{-2 s})^{-1}\prod_{p \nmid 162 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line