| L(s) = 1 | + 5-s + 4·11-s − 13-s + 6·17-s − 4·19-s − 8·23-s + 25-s + 6·29-s − 8·31-s + 10·37-s + 6·41-s − 4·43-s − 7·49-s − 10·53-s + 4·55-s + 4·59-s + 2·61-s − 65-s + 12·67-s − 16·71-s + 2·73-s − 16·79-s − 12·83-s + 6·85-s − 10·89-s − 4·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.937·41-s − 0.609·43-s − 49-s − 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.124·65-s + 1.46·67-s − 1.89·71-s + 0.234·73-s − 1.80·79-s − 1.31·83-s + 0.650·85-s − 1.05·89-s − 0.410·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−14.74783963517151, −14.57441487558369, −14.28965340058087, −13.68465090268045, −12.83621903463492, −12.61980609307166, −12.02113872376840, −11.45505741159342, −11.01316662095746, −10.15688517969858, −9.829736082188927, −9.455852583287835, −8.678170565349527, −8.176073372477673, −7.631779372474539, −6.929047183805252, −6.298514895868860, −5.916425488118716, −5.326088125587489, −4.381264013650675, −4.073224088782909, −3.259456926351649, −2.549879453325441, −1.709921129712363, −1.172813112755151, 0,
1.172813112755151, 1.709921129712363, 2.549879453325441, 3.259456926351649, 4.073224088782909, 4.381264013650675, 5.326088125587489, 5.916425488118716, 6.298514895868860, 6.929047183805252, 7.631779372474539, 8.176073372477673, 8.678170565349527, 9.455852583287835, 9.829736082188927, 10.15688517969858, 11.01316662095746, 11.45505741159342, 12.02113872376840, 12.61980609307166, 12.83621903463492, 13.68465090268045, 14.28965340058087, 14.57441487558369, 14.74783963517151
