Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | − 3-s + 0.200·5-s + 1.28·7-s + 2·9-s + 0.454·11-s − 0.692·13-s − 0.200·15-s − 0.764·17-s − 0.157·19-s − 1.28·21-s − 0.304·23-s + 0.0400·25-s − 2·27-s − 0.862·29-s + 1.06·31-s − 0.454·33-s + 0.257·35-s − 0.0270·37-s + 0.692·39-s − 0.121·41-s + 0.488·43-s + 0.400·45-s − 0.659·47-s + 0.367·49-s + 0.764·51-s − 0.320·53-s + 0.0909·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(400\) = \(2^{4} \cdot 5^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 400,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-5^{- s})^{-1}\prod_{p \nmid 40 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.9829561749\]
\[L(1, E, \mathrm{sym}^{2}) \approx 0.9415004408\]