| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 13-s − 14-s + 16-s + 6·17-s + 4·19-s − 6·23-s − 5·25-s − 26-s − 28-s + 8·31-s + 32-s + 6·34-s + 2·37-s + 4·38-s + 4·43-s − 6·46-s − 6·47-s + 49-s − 5·50-s − 52-s − 56-s − 6·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s − 0.196·26-s − 0.188·28-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s + 0.609·43-s − 0.884·46-s − 0.875·47-s + 1/7·49-s − 0.707·50-s − 0.138·52-s − 0.133·56-s − 0.781·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.46741348959738, −12.73455474111403, −12.30758670600357, −12.05739608824243, −11.56452748721013, −11.16045896974297, −10.32214287854834, −10.09258854216716, −9.642150694105774, −9.255719308402148, −8.347896414599755, −7.887583846000439, −7.683508657096078, −6.997801598984107, −6.457925942607943, −5.853352818594230, −5.683922455090071, −4.989711906096767, −4.441359263496952, −3.893225526316219, −3.355809827214018, −2.894233458805150, −2.298050792568251, −1.540033397688228, −0.9440728411154336, 0,
0.9440728411154336, 1.540033397688228, 2.298050792568251, 2.894233458805150, 3.355809827214018, 3.893225526316219, 4.441359263496952, 4.989711906096767, 5.683922455090071, 5.853352818594230, 6.457925942607943, 6.997801598984107, 7.683508657096078, 7.887583846000439, 8.347896414599755, 9.255719308402148, 9.642150694105774, 10.09258854216716, 10.32214287854834, 11.16045896974297, 11.56452748721013, 12.05739608824243, 12.30758670600357, 12.73455474111403, 13.46741348959738
