| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s − 6·13-s − 2·14-s − 16-s − 6·17-s + 4·19-s − 20-s − 4·23-s + 25-s + 6·26-s − 2·28-s − 2·31-s − 5·32-s + 6·34-s + 2·35-s − 2·37-s − 4·38-s + 3·40-s + 6·41-s − 6·43-s + 4·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s − 1.66·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.377·28-s − 0.359·31-s − 0.883·32-s + 1.02·34-s + 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s − 0.914·43-s + 0.589·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.84544833027172, −12.27596007022086, −11.67443425741417, −11.38812478292578, −10.87303210311678, −10.24749729431820, −10.01374040319076, −9.509382634855525, −9.238579996866767, −8.550908991693915, −8.362612760201660, −7.687694072185600, −7.369736360341860, −6.917377536193607, −6.368480592059009, −5.536809485304468, −5.265466662750501, −4.757420232445747, −4.343788018937577, −3.882599633811028, −2.987994715938696, −2.439949168796841, −1.854251674788537, −1.498531693893711, −0.5884192301013120, 0,
0.5884192301013120, 1.498531693893711, 1.854251674788537, 2.439949168796841, 2.987994715938696, 3.882599633811028, 4.343788018937577, 4.757420232445747, 5.265466662750501, 5.536809485304468, 6.368480592059009, 6.917377536193607, 7.369736360341860, 7.687694072185600, 8.362612760201660, 8.550908991693915, 9.238579996866767, 9.509382634855525, 10.01374040319076, 10.24749729431820, 10.87303210311678, 11.38812478292578, 11.67443425741417, 12.27596007022086, 12.84544833027172
