| L(s) = 1 | − 2-s + 4-s − 8-s + 5·11-s + 13-s + 16-s + 7·17-s − 7·19-s − 5·22-s − 2·23-s − 5·25-s − 26-s + 9·29-s − 32-s − 7·34-s + 4·37-s + 7·38-s + 4·41-s + 2·43-s + 5·44-s + 2·46-s − 3·47-s + 5·50-s + 52-s − 53-s − 9·58-s + 7·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.50·11-s + 0.277·13-s + 1/4·16-s + 1.69·17-s − 1.60·19-s − 1.06·22-s − 0.417·23-s − 25-s − 0.196·26-s + 1.67·29-s − 0.176·32-s − 1.20·34-s + 0.657·37-s + 1.13·38-s + 0.624·41-s + 0.304·43-s + 0.753·44-s + 0.294·46-s − 0.437·47-s + 0.707·50-s + 0.138·52-s − 0.137·53-s − 1.18·58-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.770247160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770247160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.43221194200092, −16.20076528770762, −15.17318136609535, −14.91889540681716, −14.12126576806863, −13.86607239431073, −12.83421975851155, −12.14727149222674, −11.99784077954634, −11.14615417994453, −10.61604895811314, −9.879170145035301, −9.536285856765503, −8.769001329008798, −8.219545803524473, −7.710701720179328, −6.860750393194667, −6.203269137993797, −5.923687154713280, −4.743499327925045, −4.001654675362855, −3.378292978943410, −2.365089644103606, −1.504999616101305, −0.7321569013557583,
0.7321569013557583, 1.504999616101305, 2.365089644103606, 3.378292978943410, 4.001654675362855, 4.743499327925045, 5.923687154713280, 6.203269137993797, 6.860750393194667, 7.710701720179328, 8.219545803524473, 8.769001329008798, 9.536285856765503, 9.879170145035301, 10.61604895811314, 11.14615417994453, 11.99784077954634, 12.14727149222674, 12.83421975851155, 13.86607239431073, 14.12126576806863, 14.91889540681716, 15.17318136609535, 16.20076528770762, 16.43221194200092
