Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.353·2-s − 0.962·3-s + 0.125·4-s − 0.0894·5-s + 0.340·6-s + 2.96·7-s − 0.0441·8-s − 0.185·9-s + 0.0316·10-s + 0.0274·11-s − 0.120·12-s − 0.938·13-s − 1.04·14-s + 0.0860·15-s + 0.0156·16-s − 1.07·17-s + 0.0654·18-s − 0.929·19-s − 0.0111·20-s − 2.85·21-s − 0.00969·22-s + 0.543·23-s + 0.0425·24-s + 0.00800·25-s + 0.331·26-s + 0.285·27-s + 0.371·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 1331000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3})
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(1331000\) = \(2^{3} \cdot 5^{3} \cdot 11^{3}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 1331000,\ (\ :1.5, 0.5),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1+5^{ -s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 110 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
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.