| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 4·7-s − 8-s + 9-s + 10-s − 3·11-s + 12-s − 13-s + 4·14-s − 15-s + 16-s − 18-s + 2·19-s − 20-s − 4·21-s + 3·22-s − 3·23-s − 24-s + 25-s + 26-s + 27-s − 4·28-s + 30-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 − 0.904·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.872·21-s + 0.639·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.97568222312927, −14.16124610901855, −13.81432196891325, −13.11972672345065, −12.71697530444119, −12.26310736109288, −11.77297756896270, −10.92659991684040, −10.54766569469598, −9.973637533451954, −9.555832230368878, −9.137930701030633, −8.479679227289839, −7.960550794389634, −7.485116346710956, −6.880838247664331, −6.539391615141298, −5.645092833036808, −5.230450990668265, −4.209695735115673, −3.594803497142470, −3.109861480047127, −2.516918150600612, −1.848150664564422, −0.7059527307948165, 0,
0.7059527307948165, 1.848150664564422, 2.516918150600612, 3.109861480047127, 3.594803497142470, 4.209695735115673, 5.230450990668265, 5.645092833036808, 6.539391615141298, 6.880838247664331, 7.485116346710956, 7.960550794389634, 8.479679227289839, 9.137930701030633, 9.555832230368878, 9.973637533451954, 10.54766569469598, 10.92659991684040, 11.77297756896270, 12.26310736109288, 12.71697530444119, 13.11972672345065, 13.81432196891325, 14.16124610901855, 14.97568222312927
