| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s + 2·11-s − 12-s + 13-s + 15-s + 16-s − 6·17-s + 18-s + 6·19-s − 20-s + 2·22-s + 4·23-s − 24-s + 25-s + 26-s − 27-s + 30-s − 4·31-s + 32-s − 2·33-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 + 0.603·11-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 + 1.37·19-s − 0.223·20-s + 0.426·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.182·30-s − 0.718·31-s + 0.176·32-s − 0.348·33-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
| \(L(1)\) |
\(\approx\) |
\(2.701217783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.701217783\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.71565800047403, −15.08694858732491, −14.85256817826492, −13.87925587262438, −13.57264937226190, −13.00895137795315, −12.38397007995170, −11.80801944238474, −11.38829367908889, −11.01146763648968, −10.34373646747689, −9.572580902210137, −9.010827705481651, −8.343466253741832, −7.549281399298008, −6.910938950185541, −6.614737179567607, −5.810956490034926, −5.132026451557764, −4.678237085214887, −3.870332676467142, −3.409056423182881, −2.472977161855563, −1.555687883815118, −0.6522805155145664,
0.6522805155145664, 1.555687883815118, 2.472977161855563, 3.409056423182881, 3.870332676467142, 4.678237085214887, 5.132026451557764, 5.810956490034926, 6.614737179567607, 6.910938950185541, 7.549281399298008, 8.343466253741832, 9.010827705481651, 9.572580902210137, 10.34373646747689, 11.01146763648968, 11.38829367908889, 11.80801944238474, 12.38397007995170, 13.00895137795315, 13.57264937226190, 13.87925587262438, 14.85256817826492, 15.08694858732491, 15.71565800047403
