Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.192·3-s − 1.07·5-s − 1.07·7-s + 0.0370·9-s − 0.986·11-s + 0.533·13-s + 0.206·15-s + 0.446·19-s + 0.207·21-s + 0.912·25-s − 0.00712·27-s + 1.07·29-s + 0.990·31-s + 0.189·33-s + 1.15·35-s + 0.866·37-s − 0.102·39-s − 0.685·41-s + 1.08·43-s − 0.0397·45-s + 1.08·47-s + 0.166·49-s − 1.18·53-s + 1.05·55-s − 0.0859·57-s + 0.688·59-s + 0.461·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5419008 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(5419008\) = \(2^{12} \cdot 3^{3} \cdot 7^{2}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 5419008,\ (\ :1.5, 0.5),\ -1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1+20\ 7^{- s}+343\ 7^{-2 s})^{-1}\prod_{p \nmid 9408 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.