Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | − 0.353·2-s + 0.769·3-s + 0.125·4-s − 0.272·6-s + 0.0539·7-s − 0.0441·8-s + 0.814·9-s + 0.0962·12-s + 0.853·13-s − 0.0190·14-s + 0.0156·16-s + 0.171·17-s − 0.288·18-s − 0.821·19-s + 0.0415·21-s − 0.0340·24-s − 2·25-s − 0.301·26-s + 1.56·27-s + 0.00674·28-s + 0.845·29-s + 1.06·31-s − 0.00552·32-s − 0.0605·34-s + 0.101·36-s − 0.622·37-s + 0.290·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(2744\) = \(2^{3} \cdot 7^{3}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | \((4,\ 2744,\ (\ :1.5, 0.5),\ 1)\) |
Euler product
\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 14 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.258906161\]
\[L(1, E, \mathrm{sym}^{3}) \approx 1.143224753\]