| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 4·11-s + 2·13-s − 15-s + 6·17-s + 8·19-s + 4·21-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s − 4·35-s + 2·37-s − 2·39-s + 2·41-s + 43-s + 45-s + 8·47-s + 9·49-s − 6·51-s − 6·53-s − 4·55-s − 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.22963928920687, −14.34302507779557, −13.82814473184367, −13.44816132965256, −12.84177214016273, −12.47179670514749, −12.09023170346718, −11.23733449064132, −10.80478037981123, −10.20135401460695, −9.800596942622946, −9.366171636748402, −8.863854204725930, −7.707048087782041, −7.604000169906983, −6.962436837692348, −6.084642699381938, −5.781741978403267, −5.421350824491338, −4.682784083174779, −3.687307674914514, −3.193910227674341, −2.776998809736673, −1.660568479447509, −0.8796128978466114, 0,
0.8796128978466114, 1.660568479447509, 2.776998809736673, 3.193910227674341, 3.687307674914514, 4.682784083174779, 5.421350824491338, 5.781741978403267, 6.084642699381938, 6.962436837692348, 7.604000169906983, 7.707048087782041, 8.863854204725930, 9.366171636748402, 9.800596942622946, 10.20135401460695, 10.80478037981123, 11.23733449064132, 12.09023170346718, 12.47179670514749, 12.84177214016273, 13.44816132965256, 13.82814473184367, 14.34302507779557, 15.22963928920687
