Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.0539·7-s − 0.411·11-s − 1.08·13-s + 0.470·17-s + 0.144·19-s + 0.543·23-s + 1.32·29-s − 1.06·31-s + 0.622·37-s − 1.00·41-s + 0.496·43-s − 0.288·47-s + 0.00291·49-s − 0.933·53-s + 0.953·59-s + 0.973·61-s + 0.218·67-s − 1.04·71-s + 0.455·73-s + 0.0222·77-s − 0.203·79-s − 0.793·83-s + 0.771·89-s + 0.0587·91-s − 0.340·97-s + 0.981·101-s + 2.81·103-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{3}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+7^{ -s})^{-1}\prod_{p \nmid 25200 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.