| L(s) = 1 | + 2·5-s − 6·13-s + 2·17-s − 25-s − 4·29-s + 12·37-s + 8·41-s − 7·49-s + 14·53-s − 12·61-s − 12·65-s − 6·73-s + 4·85-s − 10·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.66·13-s + 0.485·17-s − 1/5·25-s − 0.742·29-s + 1.97·37-s + 1.24·41-s − 49-s + 1.92·53-s − 1.53·61-s − 1.48·65-s − 0.702·73-s + 0.433·85-s − 1.05·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 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 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.37992088229884, −13.25836228730529, −12.50774208380109, −12.32477651438925, −11.61149694374500, −11.26836516376699, −10.60052056230610, −10.06211775842931, −9.765314035979117, −9.339067219636402, −8.954076727430914, −8.120175883211527, −7.632689577637104, −7.360460889714124, −6.674874877929945, −6.078749405478155, −5.642722356051794, −5.255919681844499, −4.487690376160507, −4.208291207099912, −3.293634786978708, −2.656284730549181, −2.299105620980705, −1.634610630328710, −0.8630422758102947, 0,
0.8630422758102947, 1.634610630328710, 2.299105620980705, 2.656284730549181, 3.293634786978708, 4.208291207099912, 4.487690376160507, 5.255919681844499, 5.642722356051794, 6.078749405478155, 6.674874877929945, 7.360460889714124, 7.632689577637104, 8.120175883211527, 8.954076727430914, 9.339067219636402, 9.765314035979117, 10.06211775842931, 10.60052056230610, 11.26836516376699, 11.61149694374500, 12.32477651438925, 12.50774208380109, 13.25836228730529, 13.37992088229884
