| L(s) = 1 | − 3-s − 3·5-s + 2·7-s − 2·9-s − 11-s + 4·13-s + 3·15-s + 6·17-s − 2·21-s − 3·23-s + 4·25-s + 5·27-s − 5·31-s + 33-s − 6·35-s + 37-s − 4·39-s − 10·43-s + 6·45-s − 3·49-s − 6·51-s + 6·53-s + 3·55-s − 3·59-s − 4·61-s − 4·63-s − 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.774·15-s + 1.45·17-s − 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 0.174·33-s − 1.01·35-s + 0.164·37-s − 0.640·39-s − 1.52·43-s + 0.894·45-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.404·55-s − 0.390·59-s − 0.512·61-s − 0.503·63-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15884 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.28699418932186, −15.86095522155408, −15.05986097569256, −14.72345660964076, −14.15788470198542, −13.48163350874755, −12.80119058207576, −12.02532645003594, −11.76863929590854, −11.37514187950509, −10.70571848727287, −10.33888388086388, −9.381590342757092, −8.491569754188170, −8.284063515131377, −7.665444718422954, −7.134887834158253, −6.138167532740009, −5.712386141778874, −4.993792933784728, −4.371920354855901, −3.496372528288579, −3.188608157438052, −1.868393743535220, −0.9463307051707093, 0,
0.9463307051707093, 1.868393743535220, 3.188608157438052, 3.496372528288579, 4.371920354855901, 4.993792933784728, 5.712386141778874, 6.138167532740009, 7.134887834158253, 7.665444718422954, 8.284063515131377, 8.491569754188170, 9.381590342757092, 10.33888388086388, 10.70571848727287, 11.37514187950509, 11.76863929590854, 12.02532645003594, 12.80119058207576, 13.48163350874755, 14.15788470198542, 14.72345660964076, 15.05986097569256, 15.86095522155408, 16.28699418932186
