Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 1.07·5-s + 0.657·11-s + 1.28·13-s − 0.856·17-s + 1.06·19-s − 1.30·23-s + 0.912·25-s + 0.691·29-s + 1.15·37-s + 1.05·41-s − 0.993·43-s − 1.08·53-s − 0.706·55-s + 0.900·59-s − 1.08·61-s − 1.37·65-s + 0.860·67-s + 1.04·71-s + 0.737·73-s − 0.793·83-s + 0.918·85-s + 1.01·89-s − 1.14·95-s + 0.0293·97-s − 0.390·101-s + 1.08·103-s − 0.758·107-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{12} \cdot 3^{4} \cdot 7^{4}\) |
| Sign: | $-1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{12} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 28224 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.