| L(s) = 1 | + 3·3-s + 2·5-s − 7-s + 6·9-s − 4·13-s + 6·15-s − 19-s − 3·21-s + 6·23-s − 25-s + 9·27-s + 3·29-s + 9·31-s − 2·35-s + 6·37-s − 12·39-s + 6·41-s − 6·43-s + 12·45-s − 47-s + 49-s − 13·53-s − 3·57-s − 7·59-s − 10·61-s − 6·63-s − 8·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 0.377·7-s + 2·9-s − 1.10·13-s + 1.54·15-s − 0.229·19-s − 0.654·21-s + 1.25·23-s − 1/5·25-s + 1.73·27-s + 0.557·29-s + 1.61·31-s − 0.338·35-s + 0.986·37-s − 1.92·39-s + 0.937·41-s − 0.914·43-s + 1.78·45-s − 0.145·47-s + 1/7·49-s − 1.78·53-s − 0.397·57-s − 0.911·59-s − 1.28·61-s − 0.755·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.89057428063187, −13.18823899046755, −13.00488344917423, −12.59005503819042, −11.90585777412422, −11.34621831486754, −10.55490589490554, −10.09070066947978, −9.747030555648466, −9.387997049277589, −8.907639259711825, −8.458808734734964, −7.809921916734138, −7.502695880073474, −6.891525485469414, −6.337431819038989, −5.873456620846199, −4.956944515713422, −4.559529937035668, −4.064960906465904, −3.089615606363159, −2.798570613859216, −2.562679581770165, −1.624366655339772, −1.265988669390431, 0,
1.265988669390431, 1.624366655339772, 2.562679581770165, 2.798570613859216, 3.089615606363159, 4.064960906465904, 4.559529937035668, 4.956944515713422, 5.873456620846199, 6.337431819038989, 6.891525485469414, 7.502695880073474, 7.809921916734138, 8.458808734734964, 8.907639259711825, 9.387997049277589, 9.747030555648466, 10.09070066947978, 10.55490589490554, 11.34621831486754, 11.90585777412422, 12.59005503819042, 13.00488344917423, 13.18823899046755, 13.89057428063187
