Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.353·2-s + 0.125·4-s + 0.0894·5-s + 1.07·7-s − 0.0441·8-s − 0.0316·10-s + 1.06·11-s − 0.106·13-s − 0.381·14-s + 0.0156·16-s − 0.171·17-s + 0.446·19-s + 0.0111·20-s − 0.377·22-s + 1.00·23-s + 0.00800·25-s + 0.0377·26-s + 0.134·28-s − 0.845·29-s + 1.06·31-s − 0.00552·32-s + 0.0605·34-s + 0.0965·35-s + 2.29·37-s − 0.157·38-s − 0.00395·40-s + 0.834·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3969000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(3969000\) = \(2^{3} \cdot 3^{4} \cdot 5^{3} \cdot 7^{2}\) |
| Sign: | $1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 3969000,\ (\ :1.5, 0.5),\ 1)\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1} (1-5^{- s})^{-1}(1-20\ 7^{- s}+343\ 7^{-2 s})^{-1}\prod_{p \nmid 4410 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.