Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 2-s + 2·3-s + 2.20·5-s + 2·6-s + 1.28·7-s + 2·9-s + 2.20·10-s + 2.27·11-s + 0.230·13-s + 1.28·14-s + 4.40·15-s + 16-s − 0.0588·17-s + 2·18-s + 1.57·19-s + 2.57·21-s + 2.27·22-s + 0.565·23-s + 2.64·25-s + 0.230·26-s + 27-s + 0.241·29-s + 4.40·30-s − 0.870·31-s + 32-s + 4.54·33-s − 0.0588·34-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 25775929 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(25775929\) = \(5077^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 25775929,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{2}) = (1-5077^{- s})^{-1}\prod_{p \nmid 5077 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
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.