| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s + 2·13-s − 2·15-s − 2·17-s + 4·19-s − 4·21-s − 25-s − 27-s + 6·29-s − 4·31-s + 8·35-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s + 2·45-s + 8·47-s + 9·49-s + 2·51-s − 6·53-s − 4·57-s − 12·59-s + 10·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.879979202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.879979202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−16.52184204048019, −15.78994544082122, −15.43846413693129, −14.50624606125631, −14.19934527767619, −13.62244908899773, −13.10999165973361, −12.31360296003130, −11.67045961527772, −11.34230473006151, −10.49051277856116, −10.37729283042030, −9.233589273000301, −9.017917542192548, −8.003315864282466, −7.648384051021589, −6.751781663071397, −6.089677841694109, −5.507753389428934, −4.919443632076926, −4.359506672648896, −3.378733524570217, −2.263796322482704, −1.643418835165879, −0.8603372790458047,
0.8603372790458047, 1.643418835165879, 2.263796322482704, 3.378733524570217, 4.359506672648896, 4.919443632076926, 5.507753389428934, 6.089677841694109, 6.751781663071397, 7.648384051021589, 8.003315864282466, 9.017917542192548, 9.233589273000301, 10.37729283042030, 10.49051277856116, 11.34230473006151, 11.67045961527772, 12.31360296003130, 13.10999165973361, 13.62244908899773, 14.19934527767619, 14.50624606125631, 15.43846413693129, 15.78994544082122, 16.52184204048019
