Dirichlet series
| $L(s, E, \mathrm{sym}^{2})$ = 1 | + 0.5·2-s − 0.666·3-s + 0.250·4-s + 0.200·5-s − 0.333·6-s + 2.57·7-s + 0.125·8-s + 1.11·9-s + 0.100·10-s + 0.0909·11-s − 0.166·12-s − 0.692·13-s + 1.28·14-s − 0.133·15-s + 0.0625·16-s − 0.470·17-s + 0.555·18-s + 1.57·19-s + 0.0500·20-s − 1.71·21-s + 0.0454·22-s + 0.565·23-s − 0.0833·24-s + 0.0400·25-s − 0.346·26-s − 0.185·27-s + 0.642·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 12100 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr
=\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(12100\) = \(2^{2} \cdot 5^{2} \cdot 11^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 12100,\ (1:1.0),\ 1)$ |
Euler product
\[\begin{aligned}
L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-5^{- s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 110 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}
\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{2}) \approx 3.428652360\]
\[L(1, E, \mathrm{sym}^{2}) \approx 1.757660293\]