Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | − 0.5·2-s + 0.333·3-s + 0.750·4-s − 0.200·5-s − 0.166·6-s + 0.142·7-s + 0.375·8-s + 0.111·9-s + 0.100·10-s + 0.454·11-s + 0.250·12-s − 0.692·13-s − 0.0714·14-s − 0.0666·15-s − 0.312·16-s + 1.11·17-s − 0.0555·18-s − 0.157·19-s − 0.149·20-s + 0.0476·21-s − 0.227·22-s − 23-s + 0.125·24-s + 0.239·25-s + 0.346·26-s + 0.0370·27-s + 0.107·28-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: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((3,\ 441,\ (1:1.0),\ 1)\) |
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.112873013\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.031731256\]
Euler product
\(L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 63 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line