| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 12-s + 13-s − 2·14-s − 15-s + 16-s − 4·17-s + 18-s + 2·19-s + 20-s + 2·21-s − 6·23-s − 24-s + 25-s + 26-s − 27-s − 2·28-s + 4·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 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.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−14.78375079723541, −14.12040755248241, −13.90296719379940, −13.15399362159997, −12.86122200746030, −12.42240006891714, −11.77442805351404, −11.31905786548614, −10.76936460878230, −10.27223125558096, −9.725511807958608, −9.211148421051290, −8.589814275501108, −7.819638683915587, −7.183546279024000, −6.705994037298409, −6.068553486231479, −5.841201379860992, −5.140152931004640, −4.476282780506072, −3.907151513544170, −3.316493363853210, −2.472293123759960, −1.945910408133596, −0.9904223499572070, 0,
0.9904223499572070, 1.945910408133596, 2.472293123759960, 3.316493363853210, 3.907151513544170, 4.476282780506072, 5.140152931004640, 5.841201379860992, 6.068553486231479, 6.705994037298409, 7.183546279024000, 7.819638683915587, 8.589814275501108, 9.211148421051290, 9.725511807958608, 10.27223125558096, 10.76936460878230, 11.31905786548614, 11.77442805351404, 12.42240006891714, 12.86122200746030, 13.15399362159997, 13.90296719379940, 14.12040755248241, 14.78375079723541
