Dirichlet series
| $L(s, E, \mathrm{sym}^{3})$ = 1 | + 0.0894·5-s − 0.431·7-s − 2·9-s − 0.657·11-s + 0.938·13-s − 0.856·17-s − 1.06·19-s − 1.08·23-s + 0.00800·25-s + 0.691·29-s − 0.0926·31-s − 0.0386·35-s − 1.01·37-s + 1.05·41-s + 0.624·43-s − 0.178·45-s − 0.968·47-s − 0.180·49-s − 1.08·53-s − 0.0588·55-s + 0.900·59-s + 0.495·61-s + 0.863·63-s + 0.0839·65-s − 1.02·67-s + 1.05·73-s + 0.284·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 8000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(8000\) = \(2^{6} \cdot 5^{3}\) |
| \( \varepsilon \) | = | $-1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 8000,\ (\ :1.5, 0.5),\ -1)$ |
Euler product
\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-5^{- s})^{-1}\prod_{p \nmid 40 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]
Particular Values
\[L(1/2, E, \mathrm{sym}^{3}) \approx 0\]
\[L(1, E, \mathrm{sym}^{3}) \approx 0.6440527444\]