| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 2·17-s + 4·19-s + 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s − 2·45-s − 7·49-s + 2·51-s − 2·53-s + 8·55-s + 4·57-s − 4·59-s − 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s − 49-s + 0.280·51-s − 0.274·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8428751774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8428751774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.57665942877913827434892573142, −14.73956752065291729947920771211, −13.40929752839485230085938675601, −12.32215863397544621963398389319, −11.02059068705223596291987853205, −9.656797755147918818405236990829, −8.173282349834764667604075089133, −7.26958585488936859387169650645, −5.01357758335939619070346679818, −3.14826068230388029805668446658,
3.14826068230388029805668446658, 5.01357758335939619070346679818, 7.26958585488936859387169650645, 8.173282349834764667604075089133, 9.656797755147918818405236990829, 11.02059068705223596291987853205, 12.32215863397544621963398389319, 13.40929752839485230085938675601, 14.73956752065291729947920771211, 15.57665942877913827434892573142
