Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 0.5·2-s + 2·3-s + 0.250·4-s − 0.800·5-s + 6-s − 0.857·7-s + 0.125·8-s + 2·9-s − 0.400·10-s − 0.636·11-s + 0.5·12-s + 0.0769·13-s − 0.428·14-s − 1.60·15-s + 0.0625·16-s − 0.470·17-s + 18-s + 0.894·19-s − 0.200·20-s − 1.71·21-s − 0.318·22-s − 0.304·23-s + 0.250·24-s + 1.43·25-s + 0.0384·26-s + 27-s − 0.214·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(676\) = \(2^{2} \cdot 13^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 676,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 26 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.307240091\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.895056788\]