| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s + 13-s − 14-s − 16-s + 2·17-s + 6·23-s − 5·25-s + 26-s + 28-s + 4·29-s + 5·32-s + 2·34-s + 10·37-s − 12·41-s − 4·43-s + 6·46-s − 10·47-s + 49-s − 5·50-s − 52-s + 12·53-s + 3·56-s + 4·58-s + 14·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s + 1.25·23-s − 25-s + 0.196·26-s + 0.188·28-s + 0.742·29-s + 0.883·32-s + 0.342·34-s + 1.64·37-s − 1.87·41-s − 0.609·43-s + 0.884·46-s − 1.45·47-s + 1/7·49-s − 0.707·50-s − 0.138·52-s + 1.64·53-s + 0.400·56-s + 0.525·58-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.09622705507853, −12.57546968814933, −12.04385203571866, −11.61800239410757, −11.38872617486157, −10.54805729965146, −10.14749770847562, −9.686026851805099, −9.367145694700037, −8.650500780075675, −8.430088961610430, −7.899385836491729, −7.225107090305247, −6.691047404185802, −6.332988833353971, −5.673220353257433, −5.376066157154062, −4.812943482087767, −4.309649989900497, −3.828568198683671, −3.199216780549903, −2.973237498759401, −2.211269547599218, −1.382447722568094, −0.7524230009224282, 0,
0.7524230009224282, 1.382447722568094, 2.211269547599218, 2.973237498759401, 3.199216780549903, 3.828568198683671, 4.309649989900497, 4.812943482087767, 5.376066157154062, 5.673220353257433, 6.332988833353971, 6.691047404185802, 7.225107090305247, 7.899385836491729, 8.430088961610430, 8.650500780075675, 9.367145694700037, 9.686026851805099, 10.14749770847562, 10.54805729965146, 11.38872617486157, 11.61800239410757, 12.04385203571866, 12.57546968814933, 13.09622705507853
