Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.353·2-s − 0.192·3-s + 0.125·4-s + 1.07·5-s − 0.0680·6-s + 0.0539·7-s + 0.0441·8-s + 0.0370·9-s + 0.379·10-s + 0.657·11-s − 0.0240·12-s + 0.938·13-s + 0.0190·14-s − 0.206·15-s + 0.0156·16-s − 0.171·17-s + 0.0130·18-s − 1.06·19-s + 0.134·20-s − 0.0103·21-s + 0.232·22-s − 0.00906·23-s − 0.00850·24-s + 0.912·25-s + 0.331·26-s − 0.00712·27-s + 0.00674·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+3^{ -s})^{-1}(1-7^{- s})^{-1}(1+23^{ -s})^{-1}\prod_{p \nmid 966 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
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.