| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s − 3·11-s + 12-s + 2·13-s − 14-s + 3·15-s + 16-s − 18-s + 2·19-s + 3·20-s + 21-s + 3·22-s − 24-s + 4·25-s − 2·26-s + 27-s + 28-s + 3·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.904·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.670·20-s + 0.218·21-s + 0.639·22-s − 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.869131428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.869131428\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−16.45638987977341, −15.78437590475523, −15.28647512499678, −14.72864401107122, −13.87371904925659, −13.66215062060582, −13.17731078847753, −12.32948405592873, −11.79150529307322, −10.78993835101056, −10.57056282153378, −9.831982331395306, −9.474111678746662, −8.763184419738343, −8.198403447345199, −7.720867774709609, −6.842327074285657, −6.305127559596148, −5.516371768729435, −5.023883532316550, −3.983047086306180, −2.960036324345850, −2.421751410283241, −1.695775190573781, −0.8631982657501298,
0.8631982657501298, 1.695775190573781, 2.421751410283241, 2.960036324345850, 3.983047086306180, 5.023883532316550, 5.516371768729435, 6.305127559596148, 6.842327074285657, 7.720867774709609, 8.198403447345199, 8.763184419738343, 9.474111678746662, 9.831982331395306, 10.57056282153378, 10.78993835101056, 11.79150529307322, 12.32948405592873, 13.17731078847753, 13.66215062060582, 13.87371904925659, 14.72864401107122, 15.28647512499678, 15.78437590475523, 16.45638987977341
