Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.268·5-s + 0.809·7-s − 1.06·11-s + 1.08·13-s + 0.171·17-s − 0.821·19-s + 1.30·23-s + 0.232·25-s − 1.05·29-s + 0.353·31-s − 0.217·35-s − 1.15·37-s + 0.594·41-s − 1.06·43-s − 0.744·47-s + 0.860·49-s − 0.933·53-s + 0.286·55-s − 0.503·59-s + 0.495·61-s − 0.291·65-s − 0.260·67-s − 0.666·71-s + 1.05·73-s − 0.865·77-s − 1.43·79-s + 0.654·83-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{6} \cdot 3^{4} \cdot 139^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr
=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(2^{6} \cdot 3^{4} \cdot 139^{3}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 2^{6} \cdot 3^{4} \cdot 139^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{3}) = (1+139^{ -s})^{-1}\prod_{p \nmid 10008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
Particular Values
L(1/2): not computed
L(1): not computed
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.