| L(s) = 1 | + 5-s − 3·9-s − 4·11-s − 3·13-s − 2·17-s + 6·19-s − 6·23-s + 25-s − 6·29-s + 11·31-s − 4·37-s − 8·41-s − 10·43-s − 3·45-s − 7·47-s − 2·53-s − 4·55-s + 59-s − 7·61-s − 3·65-s − 12·67-s + 4·71-s − 8·73-s + 79-s + 9·81-s − 4·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s − 0.832·13-s − 0.485·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.97·31-s − 0.657·37-s − 1.24·41-s − 1.52·43-s − 0.447·45-s − 1.02·47-s − 0.274·53-s − 0.539·55-s + 0.130·59-s − 0.896·61-s − 0.372·65-s − 1.46·67-s + 0.474·71-s − 0.936·73-s + 0.112·79-s + 81-s − 0.439·83-s − 0.216·85-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 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.09217372085248, −12.27701413304773, −11.93509087898481, −11.58398830318281, −11.19071200258012, −10.35633933922691, −10.18686936220126, −9.821167781251722, −9.269549134082073, −8.635032059987019, −8.340028709813198, −7.733461108996666, −7.475781865547689, −6.793828761338598, −6.216576331072814, −5.842137841674478, −5.292663841142954, −4.849677972183401, −4.582231821926766, −3.484041914446190, −3.180405436722431, −2.700443913715434, −2.024097250301014, −1.660839013434089, −0.5545617560096502, 0,
0.5545617560096502, 1.660839013434089, 2.024097250301014, 2.700443913715434, 3.180405436722431, 3.484041914446190, 4.582231821926766, 4.849677972183401, 5.292663841142954, 5.842137841674478, 6.216576331072814, 6.793828761338598, 7.475781865547689, 7.733461108996666, 8.340028709813198, 8.635032059987019, 9.269549134082073, 9.821167781251722, 10.18686936220126, 10.35633933922691, 11.19071200258012, 11.58398830318281, 11.93509087898481, 12.27701413304773, 13.09217372085248
