| L(s) = 1 | + 5-s − 3·9-s − 4·11-s − 4·13-s + 6·17-s + 23-s + 25-s + 29-s + 7·31-s − 5·37-s − 41-s + 5·43-s − 3·45-s − 6·53-s − 4·55-s − 4·59-s + 8·61-s − 4·65-s − 7·67-s − 8·71-s − 10·73-s − 4·79-s + 9·81-s + 5·83-s + 6·85-s − 2·89-s + 97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s − 1.10·13-s + 1.45·17-s + 0.208·23-s + 1/5·25-s + 0.185·29-s + 1.25·31-s − 0.821·37-s − 0.156·41-s + 0.762·43-s − 0.447·45-s − 0.824·53-s − 0.539·55-s − 0.520·59-s + 1.02·61-s − 0.496·65-s − 0.855·67-s − 0.949·71-s − 1.17·73-s − 0.450·79-s + 81-s + 0.548·83-s + 0.650·85-s − 0.211·89-s + 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 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.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.62661495557929, −12.36980584432993, −11.81062673953810, −11.54091813104584, −10.81568553530179, −10.42542119429437, −10.02891640446078, −9.755361443146389, −9.059906265094851, −8.671933324611976, −8.123760657606818, −7.698251774736976, −7.375581299645629, −6.751844611641099, −6.085775155676338, −5.733641969639644, −5.308112861684701, −4.839194219091002, −4.450917426298709, −3.472450991164176, −3.055891947320719, −2.668901019857728, −2.181051586103898, −1.419360113697272, −0.6694701145515984, 0,
0.6694701145515984, 1.419360113697272, 2.181051586103898, 2.668901019857728, 3.055891947320719, 3.472450991164176, 4.450917426298709, 4.839194219091002, 5.308112861684701, 5.733641969639644, 6.085775155676338, 6.751844611641099, 7.375581299645629, 7.698251774736976, 8.123760657606818, 8.671933324611976, 9.059906265094851, 9.755361443146389, 10.02891640446078, 10.42542119429437, 10.81568553530179, 11.54091813104584, 11.81062673953810, 12.36980584432993, 12.62661495557929
