| L(s) = 1 | − 5-s − 7-s − 11-s − 2·13-s + 6·17-s − 4·19-s − 4·23-s + 25-s − 6·29-s − 8·31-s + 35-s − 10·37-s − 6·41-s + 4·43-s + 49-s − 6·53-s + 55-s − 2·61-s + 2·65-s + 4·67-s − 6·73-s + 77-s + 4·79-s − 6·85-s + 14·89-s + 2·91-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.702·73-s + 0.113·77-s + 0.450·79-s − 0.650·85-s + 1.48·89-s + 0.209·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.36397119054999, −12.79304650099301, −12.55352786568174, −12.05402638312580, −11.71107029882254, −11.06023320264613, −10.49692076437355, −10.33795720889299, −9.631653541754862, −9.304161185558110, −8.695048262389293, −8.210988217987466, −7.672361406412249, −7.350354797880106, −6.864957924702233, −6.184176872081006, −5.725055824340763, −5.223507324087040, −4.759400951459119, −4.006886330294723, −3.493227744736337, −3.282763893183245, −2.309698816659668, −1.915377988344193, −1.155404893158039, 0, 0,
1.155404893158039, 1.915377988344193, 2.309698816659668, 3.282763893183245, 3.493227744736337, 4.006886330294723, 4.759400951459119, 5.223507324087040, 5.725055824340763, 6.184176872081006, 6.864957924702233, 7.350354797880106, 7.672361406412249, 8.210988217987466, 8.695048262389293, 9.304161185558110, 9.631653541754862, 10.33795720889299, 10.49692076437355, 11.06023320264613, 11.71107029882254, 12.05402638312580, 12.55352786568174, 12.79304650099301, 13.36397119054999
