Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.353·2-s + 0.192·3-s + 0.125·4-s + 0.0894·5-s + 0.0680·6-s + 0.0539·7-s + 0.0441·8-s + 0.0370·9-s + 0.0316·10-s + 0.0240·12-s + 0.0213·13-s + 0.0190·14-s + 0.0172·15-s + 0.0156·16-s + 0.171·17-s + 0.0130·18-s + 1.06·19-s + 0.0111·20-s + 0.0103·21-s + 0.00850·24-s + 0.00800·25-s + 0.00754·26-s + 0.00712·27-s + 0.00674·28-s − 0.845·29-s + 0.00608·30-s + 1.06·31-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr
=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 2730 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
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.