Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | − 0.5·2-s − 3-s + 0.750·4-s − 0.200·5-s + 0.5·6-s + 1.28·7-s + 0.375·8-s + 2·9-s + 0.100·10-s − 11-s − 0.750·12-s − 0.692·13-s − 0.642·14-s + 0.200·15-s − 0.312·16-s + 0.0588·17-s − 18-s − 0.157·19-s − 0.149·20-s − 1.28·21-s + 0.5·22-s − 0.304·23-s − 0.375·24-s + 0.239·25-s + 0.346·26-s − 2·27-s + 0.964·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(289\) = \(17^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | \((3,\ 289,\ (1:1.0),\ 1)\) |
Euler product
\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-17^{- s})^{-1}\prod_{p \nmid 17 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.7655251507\]
\[L(1, E, \mathrm{sym}^{2}) \approx 0.7850059074\]