| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s − 12-s + 13-s − 4·14-s + 16-s + 18-s − 2·19-s + 4·21-s − 6·23-s − 24-s + 26-s − 27-s − 4·28-s − 6·29-s + 2·31-s + 32-s + 36-s + 4·37-s − 2·38-s − 39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.872·21-s − 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 1/6·36-s + 0.657·37-s − 0.324·38-s − 0.160·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556060496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556060496\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.78588629074273, −12.49822812109485, −12.17822292984312, −11.60248542374896, −11.04336829219591, −10.69222650655855, −10.22498405239992, −9.644098775851814, −9.335759672161350, −8.839044775709632, −7.913460849910375, −7.697370449990019, −7.054659822066756, −6.444754922080444, −6.185110421067712, −5.843523777814723, −5.321282545413388, −4.515663285233553, −4.167066172127401, −3.619803597239223, −3.166493481941769, −2.445621039717525, −1.991339581721098, −1.073272072415186, −0.3379390063191400,
0.3379390063191400, 1.073272072415186, 1.991339581721098, 2.445621039717525, 3.166493481941769, 3.619803597239223, 4.167066172127401, 4.515663285233553, 5.321282545413388, 5.843523777814723, 6.185110421067712, 6.444754922080444, 7.054659822066756, 7.697370449990019, 7.913460849910375, 8.839044775709632, 9.335759672161350, 9.644098775851814, 10.22498405239992, 10.69222650655855, 11.04336829219591, 11.60248542374896, 12.17822292984312, 12.49822812109485, 12.78588629074273
