| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 2·13-s − 15-s − 4·17-s − 4·19-s − 4·21-s + 2·23-s + 25-s + 27-s − 10·29-s − 4·31-s + 4·35-s − 8·37-s − 2·39-s − 10·41-s + 12·43-s − 45-s − 8·47-s + 9·49-s − 4·51-s − 4·53-s − 4·57-s − 12·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.676·35-s − 1.31·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.560·51-s − 0.549·53-s − 0.529·57-s − 1.56·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.32257297880052, −14.87317232747460, −14.51820300012602, −13.52113284522877, −13.39184250988498, −12.75340256527411, −12.44437672215822, −11.83238725862176, −10.98740003576144, −10.69570721090351, −10.00476212506884, −9.382674745663930, −9.072686790922207, −8.584321305227717, −7.778327662872239, −7.226135579778713, −6.808557396859654, −6.246277250889074, −5.548828636900456, −4.749733165416650, −4.106113016127192, −3.505361319778284, −3.049029139312101, −2.272468934807333, −1.595317819537516, 0, 0,
1.595317819537516, 2.272468934807333, 3.049029139312101, 3.505361319778284, 4.106113016127192, 4.749733165416650, 5.548828636900456, 6.246277250889074, 6.808557396859654, 7.226135579778713, 7.778327662872239, 8.584321305227717, 9.072686790922207, 9.382674745663930, 10.00476212506884, 10.69570721090351, 10.98740003576144, 11.83238725862176, 12.44437672215822, 12.75340256527411, 13.39184250988498, 13.52113284522877, 14.51820300012602, 14.87317232747460, 15.32257297880052
