Dirichlet series
| $L(s, E, \mathrm{sym}^{4})$ = 1 | + 0.250·2-s + 0.111·3-s + 0.0625·4-s + 0.0400·5-s + 0.0277·6-s + 3.04·7-s + 0.0156·8-s + 0.197·9-s + 0.0100·10-s + 0.00826·11-s + 0.00694·12-s + 0.171·13-s + 0.760·14-s + 0.00444·15-s + 0.00390·16-s − 0.307·17-s + 0.0493·18-s − 0.0858·19-s + 0.00250·20-s + 0.337·21-s + 0.00206·22-s − 1.24·23-s + 0.00173·24-s + 0.00160·25-s + 0.0428·26-s − 0.142·27-s + 0.190·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr
=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{4})
\end{aligned}
\]
Invariants
| \( d \) | = | \(5\) |
| \( N \) | = | \(2^{4} \cdot 5^{4} \cdot 11^{4}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(5,\ 2^{4} \cdot 5^{4} \cdot 11^{4} ,\ ( 0 : 2.0, 1.0 ),\ 1 )$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-5^{- s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 110 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
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.