| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 2·19-s − 20-s − 4·21-s + 24-s + 25-s − 26-s − 27-s + 4·28-s − 10·29-s − 30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.21914668054602, −12.69965552000878, −11.97542428995851, −11.78341098531582, −11.40141884462393, −10.89937541886909, −10.67187893541281, −9.986940095175589, −9.491939120789317, −9.061365923937151, −8.398140580576051, −7.961139330857641, −7.662451439674321, −7.320643490842824, −6.390504113973633, −6.280742116960406, −5.320850346611984, −5.121065480822105, −4.599670299656358, −3.795831844848721, −3.475717092495755, −2.539749455549436, −1.918659895742893, −1.354937123836813, −0.8530629148565636, 0,
0.8530629148565636, 1.354937123836813, 1.918659895742893, 2.539749455549436, 3.475717092495755, 3.795831844848721, 4.599670299656358, 5.121065480822105, 5.320850346611984, 6.280742116960406, 6.390504113973633, 7.320643490842824, 7.662451439674321, 7.961139330857641, 8.398140580576051, 9.061365923937151, 9.491939120789317, 9.986940095175589, 10.67187893541281, 10.89937541886909, 11.40141884462393, 11.78341098531582, 11.97542428995851, 12.69965552000878, 13.21914668054602
