| L(s) = 1 | − 5-s + 7-s + 11-s − 2·13-s + 3·17-s + 7·19-s + 3·23-s + 25-s − 3·29-s − 4·31-s − 35-s + 10·37-s + 7·43-s + 49-s + 9·53-s − 55-s + 3·59-s + 61-s + 2·65-s − 2·67-s + 12·71-s − 4·73-s + 77-s + 2·79-s − 3·83-s − 3·85-s − 9·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.301·11-s − 0.554·13-s + 0.727·17-s + 1.60·19-s + 0.625·23-s + 1/5·25-s − 0.557·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s + 1.06·43-s + 1/7·49-s + 1.23·53-s − 0.134·55-s + 0.390·59-s + 0.128·61-s + 0.248·65-s − 0.244·67-s + 1.42·71-s − 0.468·73-s + 0.113·77-s + 0.225·79-s − 0.329·83-s − 0.325·85-s − 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.253530484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.253530484\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−12.81150250183318, −12.54835813595035, −11.95651077158838, −11.54102616643920, −11.23369463660793, −10.76315570792522, −10.01226539898315, −9.747643497828411, −9.213641548460280, −8.800213311185452, −8.144808702813028, −7.642922861459870, −7.327810141947743, −7.009771773565087, −6.118459607565998, −5.727521541107897, −5.117795414004427, −4.841273174239816, −3.919803202964879, −3.802355617649364, −2.924761064549586, −2.591261124330657, −1.733243971214552, −1.039200406334401, −0.5904471057704773,
0.5904471057704773, 1.039200406334401, 1.733243971214552, 2.591261124330657, 2.924761064549586, 3.802355617649364, 3.919803202964879, 4.841273174239816, 5.117795414004427, 5.727521541107897, 6.118459607565998, 7.009771773565087, 7.327810141947743, 7.642922861459870, 8.144808702813028, 8.800213311185452, 9.213641548460280, 9.747643497828411, 10.01226539898315, 10.76315570792522, 11.23369463660793, 11.54102616643920, 11.95651077158838, 12.54835813595035, 12.81150250183318
