Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 0.333·3-s + 2.20·5-s + 0.142·7-s − 0.222·9-s − 11-s − 13-s + 0.733·15-s − 0.764·17-s − 0.789·19-s + 0.0476·21-s + 1.78·23-s + 2.64·25-s + 0.814·27-s − 0.862·29-s − 0.483·31-s − 0.333·33-s + 0.314·35-s − 0.0270·37-s − 0.333·39-s − 0.902·41-s + 0.488·43-s − 0.488·45-s − 0.659·47-s + 0.0204·49-s − 0.254·51-s + 0.886·53-s − 2.20·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(784\) = \(2^{4} \cdot 7^{2}\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((3,\ 784,\ (1:1.0),\ 1)\) |
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.764257940\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.435200539\]
Euler product
\(L(s, E, \mathrm{sym}^{2}) = (1-7^{- s})^{-1}\prod_{p \nmid 56 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line