| L(s) = 1 | − 3·9-s − 4·11-s + 4·17-s + 2·19-s − 4·23-s − 5·25-s − 10·29-s + 6·31-s + 37-s + 6·41-s − 4·43-s + 12·47-s + 6·53-s − 2·59-s + 12·61-s + 12·67-s − 8·71-s − 2·73-s − 12·79-s + 9·81-s − 16·83-s − 12·89-s + 8·97-s + 12·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s − 1.20·11-s + 0.970·17-s + 0.458·19-s − 0.834·23-s − 25-s − 1.85·29-s + 1.07·31-s + 0.164·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s + 0.824·53-s − 0.260·59-s + 1.53·61-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 1.35·79-s + 81-s − 1.75·83-s − 1.27·89-s + 0.812·97-s + 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−13.79225434795035, −13.47402642484270, −12.84587874635087, −12.42533251786329, −11.83226306658983, −11.40325502191824, −11.08817275623941, −10.29772280646059, −9.988641091233358, −9.569199099117298, −8.862080953363255, −8.317342033141410, −7.973259438914570, −7.411971976591129, −7.073462649293970, −5.996879292304070, −5.723400510101109, −5.500439013382416, −4.748680140036517, −3.954233093936327, −3.587288311159965, −2.701125448525109, −2.491167922436032, −1.653340035383132, −0.7191090365428717, 0,
0.7191090365428717, 1.653340035383132, 2.491167922436032, 2.701125448525109, 3.587288311159965, 3.954233093936327, 4.748680140036517, 5.500439013382416, 5.723400510101109, 5.996879292304070, 7.073462649293970, 7.411971976591129, 7.973259438914570, 8.317342033141410, 8.862080953363255, 9.569199099117298, 9.988641091233358, 10.29772280646059, 11.08817275623941, 11.40325502191824, 11.83226306658983, 12.42533251786329, 12.84587874635087, 13.47402642484270, 13.79225434795035
