| L(s) = 1 | + 3-s + 3·5-s + 9-s − 11-s − 13-s + 3·15-s + 3·17-s − 19-s − 5·23-s + 4·25-s + 27-s − 3·29-s − 2·31-s − 33-s − 5·37-s − 39-s + 10·41-s − 5·43-s + 3·45-s + 3·51-s − 2·53-s − 3·55-s − 57-s + 3·61-s − 3·65-s − 8·67-s − 5·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.774·15-s + 0.727·17-s − 0.229·19-s − 1.04·23-s + 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.359·31-s − 0.174·33-s − 0.821·37-s − 0.160·39-s + 1.56·41-s − 0.762·43-s + 0.447·45-s + 0.420·51-s − 0.274·53-s − 0.404·55-s − 0.132·57-s + 0.384·61-s − 0.372·65-s − 0.977·67-s − 0.601·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.12438704643555, −12.70181345110583, −12.36764643967250, −11.78986916809055, −11.16187618953318, −10.63714622576882, −10.21715092773453, −9.788182422614231, −9.472643614391546, −9.020939598566435, −8.431308017118872, −7.966828744836664, −7.498836948631337, −6.976202306077478, −6.387806557612826, −5.867062496277848, −5.548964269515064, −4.976441690588132, −4.399220781289693, −3.711389908351128, −3.264911668732717, −2.535454707421533, −2.114952259961356, −1.686053297865296, −0.9614594301017618, 0,
0.9614594301017618, 1.686053297865296, 2.114952259961356, 2.535454707421533, 3.264911668732717, 3.711389908351128, 4.399220781289693, 4.976441690588132, 5.548964269515064, 5.867062496277848, 6.387806557612826, 6.976202306077478, 7.498836948631337, 7.966828744836664, 8.431308017118872, 9.020939598566435, 9.472643614391546, 9.788182422614231, 10.21715092773453, 10.63714622576882, 11.16187618953318, 11.78986916809055, 12.36764643967250, 12.70181345110583, 13.12438704643555
