Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 0.5·2-s − 0.666·3-s + 0.250·4-s + 0.800·5-s − 0.333·6-s − 0.857·7-s + 0.125·8-s + 1.11·9-s + 0.400·10-s + 2.27·11-s − 0.166·12-s + 0.0769·13-s − 0.428·14-s − 0.533·15-s + 0.0625·16-s − 0.470·17-s + 0.555·18-s − 0.789·19-s + 0.200·20-s + 0.571·21-s + 1.13·22-s − 23-s − 0.0833·24-s − 0.160·25-s + 0.0384·26-s − 0.185·27-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
\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.528389493\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.300523850\]