| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 4·11-s + 12-s + 2·13-s − 14-s − 2·15-s + 16-s + 17-s − 18-s − 2·20-s + 21-s − 4·22-s + 8·23-s − 24-s − 25-s − 2·26-s + 27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.00443947318292, −12.56861059462836, −11.96606146798970, −11.49278790297532, −11.30035308448134, −10.87149963100034, −10.18630651791194, −9.669234294893351, −9.318683255993735, −8.692426839557266, −8.533773459172315, −7.979268379736055, −7.527445647679369, −6.989595566525962, −6.699110282270199, −6.111137050514414, −5.301005451892125, −4.941730042596068, −4.060498117690828, −3.788536537843539, −3.323764248575686, −2.708586853665008, −1.902272053492914, −1.399196732026127, −0.8883123608397671, 0,
0.8883123608397671, 1.399196732026127, 1.902272053492914, 2.708586853665008, 3.323764248575686, 3.788536537843539, 4.060498117690828, 4.941730042596068, 5.301005451892125, 6.111137050514414, 6.699110282270199, 6.989595566525962, 7.527445647679369, 7.979268379736055, 8.533773459172315, 8.692426839557266, 9.318683255993735, 9.669234294893351, 10.18630651791194, 10.87149963100034, 11.30035308448134, 11.49278790297532, 11.96606146798970, 12.56861059462836, 13.00443947318292
