Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 2-s − 0.666·3-s − 0.800·5-s − 0.666·6-s − 0.428·7-s + 1.11·9-s − 0.800·10-s + 0.0909·11-s + 0.230·13-s − 0.428·14-s + 0.533·15-s + 16-s − 0.764·17-s + 1.11·18-s − 19-s + 0.285·21-s + 0.0909·22-s − 0.956·23-s + 1.43·25-s + 0.230·26-s − 0.185·27-s − 29-s + 0.533·30-s + 0.580·31-s + 32-s − 0.0606·33-s − 0.764·34-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(121\) = \(11^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 121,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{2}) = (1-11^{- s})^{-1}\prod_{p \nmid 11 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.8933960461\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.057599244\]