| 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 + 18-s + 4·19-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 + 0.235·18-s + 0.917·19-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 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.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 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.g |
| 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
−16.62048109561241, −15.88156430921023, −15.43514565895425, −14.95428348508706, −14.14241003979634, −13.81493397859807, −13.32636717100551, −12.91015807959009, −12.16352991029155, −11.70841581683022, −10.80215341640397, −10.07399157467546, −9.898289221739233, −9.223107131361027, −8.304421690524827, −7.725934908411055, −7.199770262085365, −6.376090039530376, −5.667880567335022, −5.299651836061111, −4.452422896258439, −3.607476335167343, −2.937332237282683, −2.230758269600829, −1.643368690332513, 0,
1.643368690332513, 2.230758269600829, 2.937332237282683, 3.607476335167343, 4.452422896258439, 5.299651836061111, 5.667880567335022, 6.376090039530376, 7.199770262085365, 7.725934908411055, 8.304421690524827, 9.223107131361027, 9.898289221739233, 10.07399157467546, 10.80215341640397, 11.70841581683022, 12.16352991029155, 12.91015807959009, 13.32636717100551, 13.81493397859807, 14.14241003979634, 14.95428348508706, 15.43514565895425, 15.88156430921023, 16.62048109561241
