| L(s) = 1 | + 3-s − 7-s + 9-s + 2·13-s − 4·17-s + 6·19-s − 21-s − 4·23-s − 5·25-s + 27-s + 2·29-s + 10·31-s − 6·37-s + 2·39-s + 12·41-s + 12·43-s − 6·47-s + 49-s − 4·51-s − 6·53-s + 6·57-s − 6·61-s − 63-s − 4·67-s − 4·69-s − 8·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.218·21-s − 0.834·23-s − 25-s + 0.192·27-s + 0.371·29-s + 1.79·31-s − 0.986·37-s + 0.320·39-s + 1.87·41-s + 1.82·43-s − 0.875·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s + 0.794·57-s − 0.768·61-s − 0.125·63-s − 0.488·67-s − 0.481·69-s − 0.949·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.05806390726683, −13.77353613022113, −13.42239588045596, −12.73721768393630, −12.34433695636219, −11.64436971608670, −11.41002764329668, −10.62809896478593, −10.14567652666659, −9.722328587379132, −9.077126071770785, −8.857133272326631, −8.026300998880212, −7.728329316940356, −7.182219648222556, −6.437800691482388, −6.054457347701341, −5.527897759146718, −4.578063664788365, −4.274997635628298, −3.614877689687180, −2.905720076745494, −2.536970100223466, −1.650548504408033, −1.007827490145686, 0,
1.007827490145686, 1.650548504408033, 2.536970100223466, 2.905720076745494, 3.614877689687180, 4.274997635628298, 4.578063664788365, 5.527897759146718, 6.054457347701341, 6.437800691482388, 7.182219648222556, 7.728329316940356, 8.026300998880212, 8.857133272326631, 9.077126071770785, 9.722328587379132, 10.14567652666659, 10.62809896478593, 11.41002764329668, 11.64436971608670, 12.34433695636219, 12.73721768393630, 13.42239588045596, 13.77353613022113, 14.05806390726683
