Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 1.73·3-s − 2.14·5-s − 0.431·7-s + 9-s − 2.30·11-s + 0.853·13-s + 3.71·15-s − 16-s + 1.02·17-s − 0.929·19-s + 0.748·21-s + 0.543·23-s + 1.96·25-s + 0.845·29-s + 0.672·31-s + 3.98·33-s + 0.927·35-s − 1.47·39-s + 0.624·43-s − 2.14·45-s + 0.363·47-s + 1.73·48-s − 0.180·49-s − 1.77·51-s + 0.583·53-s + 4.94·55-s + 1.61·57-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(5077^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr
=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(5077^{3}\) |
| \( \varepsilon \) | = | $-1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 5077^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{3}) = (1+5077^{ -s})^{-1}\prod_{p \nmid 5077 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
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.