| L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s + 2·13-s − 17-s − 4·19-s − 21-s + 8·23-s − 27-s + 6·29-s + 4·33-s + 2·37-s − 2·39-s + 10·41-s − 4·43-s + 49-s + 51-s − 6·53-s + 4·57-s + 4·59-s + 6·61-s + 63-s − 12·67-s − 8·69-s + 8·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.242·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 1/7·49-s + 0.140·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s + 0.768·61-s + 0.125·63-s − 1.46·67-s − 0.963·69-s + 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.51691641159679, −13.01055504887555, −12.80631756170055, −12.23366444050223, −11.66303749369169, −11.01112717341951, −10.87684788113163, −10.51272835069732, −9.811265563584104, −9.345119140234102, −8.677208268400193, −8.273054297033438, −7.849667931860035, −7.169552036837435, −6.713930998665579, −6.259152039932048, −5.502141508828017, −5.298757073107374, −4.444962140690067, −4.385063496683856, −3.389704094512804, −2.762543221055064, −2.307505784698320, −1.412027178066493, −0.8323578324667972, 0,
0.8323578324667972, 1.412027178066493, 2.307505784698320, 2.762543221055064, 3.389704094512804, 4.385063496683856, 4.444962140690067, 5.298757073107374, 5.502141508828017, 6.259152039932048, 6.713930998665579, 7.169552036837435, 7.849667931860035, 8.273054297033438, 8.677208268400193, 9.345119140234102, 9.811265563584104, 10.51272835069732, 10.87684788113163, 11.01112717341951, 11.66303749369169, 12.23366444050223, 12.80631756170055, 13.01055504887555, 13.51691641159679
