| L(s) = 1 | − 3-s + 5-s − 2·9-s − 3·11-s + 2·13-s − 15-s − 3·17-s + 19-s − 6·23-s + 25-s + 5·27-s − 6·29-s − 5·31-s + 3·33-s − 7·37-s − 2·39-s − 6·41-s + 43-s − 2·45-s − 3·47-s + 3·51-s − 9·53-s − 3·55-s − 57-s − 6·59-s + 8·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.904·11-s + 0.554·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 1.11·29-s − 0.898·31-s + 0.522·33-s − 1.15·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s − 0.298·45-s − 0.437·47-s + 0.420·51-s − 1.23·53-s − 0.404·55-s − 0.132·57-s − 0.781·59-s + 1.02·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−12.92074758562006, −12.34154896661766, −12.04313332073197, −11.29763377774385, −11.18350247914015, −10.61126691503962, −10.33068329136007, −9.608791370913008, −9.364241724272978, −8.590597492823951, −8.404201403655141, −7.831386213519442, −7.255408200559774, −6.726269840898351, −6.256747313418398, −5.730249377854872, −5.459681695316508, −4.961606196406707, −4.408063705790960, −3.652519445445070, −3.278911326423264, −2.596396824048998, −1.923334474773926, −1.628387350449971, −0.5232424691804819, 0,
0.5232424691804819, 1.628387350449971, 1.923334474773926, 2.596396824048998, 3.278911326423264, 3.652519445445070, 4.408063705790960, 4.961606196406707, 5.459681695316508, 5.730249377854872, 6.256747313418398, 6.726269840898351, 7.255408200559774, 7.831386213519442, 8.404201403655141, 8.590597492823951, 9.364241724272978, 9.608791370913008, 10.33068329136007, 10.61126691503962, 11.18350247914015, 11.29763377774385, 12.04313332073197, 12.34154896661766, 12.92074758562006
