Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 2-s + 0.333·3-s − 0.200·5-s + 0.333·6-s + 7-s + 0.111·9-s − 0.200·10-s − 0.636·11-s − 0.923·13-s + 14-s − 0.0666·15-s + 16-s − 17-s + 0.111·18-s − 0.947·19-s + 0.333·21-s − 0.636·22-s − 23-s + 0.239·25-s − 0.923·26-s + 0.0370·27-s − 0.448·29-s − 0.0666·30-s + 1.61·31-s + 32-s − 0.212·33-s − 34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(441\) = \(3^{2} \cdot 7^{2}\) |
| Sign: | $1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((3,\ 441,\ (1:1.0),\ 1)\) |
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.797737269\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.621788676\]
Euler product
\(L(s, E, \mathrm{sym}^{2}) = (1-2\ 2^{- s}+4\ 2^{-2 s}-8 \ 2^{-3 s})^{-1}(1-3^{- s})^{-1}(1-7\ 7^{- s})^{-1}\prod_{p \nmid 9408 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line