| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 13-s − 16-s + 4·17-s − 2·19-s − 20-s − 9·23-s + 25-s − 26-s + 5·29-s − 5·32-s − 4·34-s + 8·37-s + 2·38-s + 3·40-s − 9·41-s − 43-s + 9·46-s + 4·47-s − 50-s − 52-s − 6·53-s − 5·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 0.277·13-s − 1/4·16-s + 0.970·17-s − 0.458·19-s − 0.223·20-s − 1.87·23-s + 1/5·25-s − 0.196·26-s + 0.928·29-s − 0.883·32-s − 0.685·34-s + 1.31·37-s + 0.324·38-s + 0.474·40-s − 1.40·41-s − 0.152·43-s + 1.32·46-s + 0.583·47-s − 0.141·50-s − 0.138·52-s − 0.824·53-s − 0.656·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.57879333358129, −14.81496141956468, −14.29058817088439, −13.94625898964194, −13.34946240545307, −12.91360485067702, −12.19251194845566, −11.77461971201900, −10.99105874742709, −10.36121344349111, −9.931254162065093, −9.715683259370870, −8.773547538939386, −8.528949073156165, −7.817321309307489, −7.481047860313140, −6.478793130489757, −6.070060843398539, −5.349723137549808, −4.698103857035280, −4.064047160025933, −3.422434267054708, −2.461448559691360, −1.690184321465364, −0.9835456495132409, 0,
0.9835456495132409, 1.690184321465364, 2.461448559691360, 3.422434267054708, 4.064047160025933, 4.698103857035280, 5.349723137549808, 6.070060843398539, 6.478793130489757, 7.481047860313140, 7.817321309307489, 8.528949073156165, 8.773547538939386, 9.715683259370870, 9.931254162065093, 10.36121344349111, 10.99105874742709, 11.77461971201900, 12.19251194845566, 12.91360485067702, 13.34946240545307, 13.94625898964194, 14.29058817088439, 14.81496141956468, 15.57879333358129
