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 − 0.0441·8-s + 0.814·9-s + 2.30·11-s + 0.0962·12-s − 0.938·13-s + 0.0156·16-s − 0.0142·17-s − 0.288·18-s − 1.06·19-s − 0.814·22-s − 0.0340·24-s + 0.331·26-s + 1.56·27-s − 1.06·31-s − 0.00552·32-s + 1.77·33-s + 0.00504·34-s + 0.101·36-s − 1.03·37-s + 0.375·38-s − 0.722·39-s + 1.05·41-s + 0.624·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 5^{4} \cdot 7^{4} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2^{3} \cdot 5^{4} \cdot 7^{4} \cdot 17^{3}\) |
| Sign: | $-1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 2^{3} \cdot 5^{4} \cdot 7^{4} \cdot 17^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\) |
Particular Values
L(1/2): not computed
L(1): not computed
Euler product
\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1} (1+17^{ -s})^{-1}\prod_{p \nmid 41650 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)
Imaginary part of the first few zeros on the critical line
Zeros not available.
Graph of the $Z$-function along the critical line
Plot not available.