| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s + 6·13-s + 16-s + 2·17-s − 19-s + 20-s − 22-s + 23-s − 4·25-s − 6·26-s − 4·29-s − 32-s − 2·34-s + 8·37-s + 38-s − 40-s − 12·41-s − 7·43-s + 44-s − 46-s − 9·47-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.229·19-s + 0.223·20-s − 0.213·22-s + 0.208·23-s − 4/5·25-s − 1.17·26-s − 0.742·29-s − 0.176·32-s − 0.342·34-s + 1.31·37-s + 0.162·38-s − 0.158·40-s − 1.87·41-s − 1.06·43-s + 0.150·44-s − 0.147·46-s − 1.31·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−16.37758682528634, −15.71422681151811, −15.07349854964612, −14.73476545576717, −13.82743517594786, −13.44134752552782, −12.97576559695207, −12.15641034576415, −11.51032352410041, −11.17446404351741, −10.49199899418299, −9.895630046628661, −9.427344435654017, −8.800924634398014, −8.164685842010621, −7.840203829050026, −6.724779237227708, −6.514167006817730, −5.734311033024023, −5.208070389060467, −4.088755997700724, −3.541099564268997, −2.746571170275859, −1.669800222340834, −1.305047778748699, 0,
1.305047778748699, 1.669800222340834, 2.746571170275859, 3.541099564268997, 4.088755997700724, 5.208070389060467, 5.734311033024023, 6.514167006817730, 6.724779237227708, 7.840203829050026, 8.164685842010621, 8.800924634398014, 9.427344435654017, 9.895630046628661, 10.49199899418299, 11.17446404351741, 11.51032352410041, 12.15641034576415, 12.97576559695207, 13.44134752552782, 13.82743517594786, 14.73476545576717, 15.07349854964612, 15.71422681151811, 16.37758682528634
