Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.353·2-s − 1.73·3-s + 0.125·4-s + 0.268·5-s − 0.612·6-s + 0.809·7-s + 0.0441·8-s + 9-s + 0.0948·10-s + 0.986·11-s − 0.216·12-s + 0.106·13-s + 0.286·14-s − 0.464·15-s + 0.0156·16-s + 0.171·17-s + 0.353·18-s + 0.0335·20-s − 1.40·21-s + 0.348·22-s + 0.761·23-s − 0.0765·24-s + 0.232·25-s + 0.0377·26-s + 0.101·28-s − 0.845·29-s − 0.164·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3944312 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(3944312\) = \(2^{3} \cdot 79^{3}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 3944312,\ (\ :1.5, 0.5),\ -1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1-79^{- s})^{-1}\prod_{p \nmid 158 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line
Zeros not available.
Graph of the $Z$-function along the critical line
Plot not available.