Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 2-s + 2·3-s − 0.200·5-s + 2·6-s − 0.857·7-s + 2·9-s − 0.200·10-s + 1.27·11-s − 0.692·13-s − 0.857·14-s − 0.400·15-s + 16-s − 17-s + 2·18-s − 19-s − 1.71·21-s + 1.27·22-s − 0.826·23-s + 0.239·25-s − 0.692·26-s + 27-s + 0.241·29-s − 0.400·30-s − 0.483·31-s + 32-s + 2.54·33-s − 34-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1369 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(1369\) = \(37^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 1369,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{2}) = (1-37^{- s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 3.513394223\]
\[L(1, E, \mathrm{sym}^{2}) \approx 2.492262044\]