| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 4·13-s − 15-s − 4·19-s − 4·21-s + 6·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s + 4·35-s + 10·37-s + 4·39-s + 41-s − 10·43-s + 45-s + 12·47-s + 9·49-s − 12·53-s − 4·55-s + 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 0.258·15-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s + 1.64·37-s + 0.640·39-s + 0.156·41-s − 1.52·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s − 1.64·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.05604235181818, −14.54763245184758, −14.19440769596502, −13.35687219263170, −12.99744535494255, −12.45708540958906, −11.95583900641139, −11.20084397254707, −10.95136884367019, −10.46200149210683, −9.932651521440627, −9.230968690789580, −8.651600009303105, −8.003227283204407, −7.562853061273937, −7.067686366306988, −6.296517624264513, −5.648326392902298, −5.032179026951002, −4.802634231756196, −4.241500294205388, −3.059744574889370, −2.419291015721394, −1.841783799720731, −1.009762837411425, 0,
1.009762837411425, 1.841783799720731, 2.419291015721394, 3.059744574889370, 4.241500294205388, 4.802634231756196, 5.032179026951002, 5.648326392902298, 6.296517624264513, 7.067686366306988, 7.562853061273937, 8.003227283204407, 8.651600009303105, 9.230968690789580, 9.932651521440627, 10.46200149210683, 10.95136884367019, 11.20084397254707, 11.95583900641139, 12.45708540958906, 12.99744535494255, 13.35687219263170, 14.19440769596502, 14.54763245184758, 15.05604235181818
