| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 4·11-s + 12-s − 13-s + 15-s + 16-s + 6·17-s − 18-s + 8·19-s + 20-s − 4·22-s + 8·23-s − 24-s + 25-s + 26-s + 27-s + 2·29-s − 30-s − 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.220318398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220318398\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 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.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−15.73235419961608, −15.11905283939625, −14.64636162501572, −14.11553930239573, −13.76514163410376, −12.91080460457499, −12.39766331067414, −11.75808210223235, −11.36058669209388, −10.51631024111156, −9.980297160731634, −9.468165315126614, −9.049197778141554, −8.607131429979853, −7.620998586602981, −7.313940157565574, −6.827364440025451, −5.841614065256528, −5.424741748224596, −4.577787726404659, −3.475957271380322, −3.230311773526608, −2.307893640503235, −1.301489986399174, −0.9705678323177808,
0.9705678323177808, 1.301489986399174, 2.307893640503235, 3.230311773526608, 3.475957271380322, 4.577787726404659, 5.424741748224596, 5.841614065256528, 6.827364440025451, 7.313940157565574, 7.620998586602981, 8.607131429979853, 9.049197778141554, 9.468165315126614, 9.980297160731634, 10.51631024111156, 11.36058669209388, 11.75808210223235, 12.39766331067414, 12.91080460457499, 13.76514163410376, 14.11553930239573, 14.64636162501572, 15.11905283939625, 15.73235419961608
