| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 11-s + 2·14-s + 16-s − 17-s + 2·19-s − 20-s + 22-s + 3·23-s + 25-s + 2·28-s − 3·29-s + 3·31-s + 32-s − 34-s − 2·35-s − 7·37-s + 2·38-s − 40-s − 2·41-s − 7·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.377·28-s − 0.557·29-s + 0.538·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s − 1.15·37-s + 0.324·38-s − 0.158·40-s − 0.312·41-s − 1.06·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.10765284606663, −12.48839058171025, −12.18776326903272, −11.66390317406128, −11.26926060763503, −10.98995777820279, −10.41769921488054, −9.848275706127677, −9.393151255003771, −8.697675825945114, −8.358811004159835, −7.900034901579780, −7.282416136022765, −6.908963653270983, −6.507616920064324, −5.766214373113078, −5.264625097602838, −4.947831623363288, −4.333097806677594, −3.906690709079114, −3.301671373298566, −2.857674964941202, −2.082599460912342, −1.560427107381374, −0.9424008303799133, 0,
0.9424008303799133, 1.560427107381374, 2.082599460912342, 2.857674964941202, 3.301671373298566, 3.906690709079114, 4.333097806677594, 4.947831623363288, 5.264625097602838, 5.766214373113078, 6.507616920064324, 6.908963653270983, 7.282416136022765, 7.900034901579780, 8.358811004159835, 8.697675825945114, 9.393151255003771, 9.848275706127677, 10.41769921488054, 10.98995777820279, 11.26926060763503, 11.66390317406128, 12.18776326903272, 12.48839058171025, 13.10765284606663
