| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 2·7-s − 8-s + 9-s + 2·10-s − 11-s + 12-s + 2·13-s + 2·14-s − 2·15-s + 16-s − 2·17-s − 18-s − 2·20-s − 2·21-s + 22-s − 2·23-s − 24-s − 25-s − 2·26-s + 27-s − 2·28-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s − 0.436·21-s + 0.213·22-s − 0.417·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106384748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106384748\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.47661705658023, −13.96896773675812, −13.26949525501553, −12.82319482503630, −12.47932453700526, −11.71661488876298, −11.23003386310451, −10.87876295314989, −10.19658672657108, −9.548471805563999, −9.351523009825596, −8.635899844878110, −8.116795316915567, −7.707541255899814, −7.302913040569282, −6.450804879165889, −6.200053653517215, −5.397135890985754, −4.430079782056856, −3.996651332844727, −3.414133287341617, −2.702922399319136, −2.225319239946400, −1.192561941761576, −0.4261506661419606,
0.4261506661419606, 1.192561941761576, 2.225319239946400, 2.702922399319136, 3.414133287341617, 3.996651332844727, 4.430079782056856, 5.397135890985754, 6.200053653517215, 6.450804879165889, 7.302913040569282, 7.707541255899814, 8.116795316915567, 8.635899844878110, 9.351523009825596, 9.548471805563999, 10.19658672657108, 10.87876295314989, 11.23003386310451, 11.71661488876298, 12.47932453700526, 12.82319482503630, 13.26949525501553, 13.96896773675812, 14.47661705658023
