| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 5·11-s + 16-s − 8·19-s − 20-s − 5·22-s − 4·23-s − 4·25-s − 5·29-s + 3·31-s − 32-s + 4·37-s + 8·38-s + 40-s + 2·43-s + 5·44-s + 4·46-s − 6·47-s + 4·50-s + 9·53-s − 5·55-s + 5·58-s − 11·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.50·11-s + 1/4·16-s − 1.83·19-s − 0.223·20-s − 1.06·22-s − 0.834·23-s − 4/5·25-s − 0.928·29-s + 0.538·31-s − 0.176·32-s + 0.657·37-s + 1.29·38-s + 0.158·40-s + 0.304·43-s + 0.753·44-s + 0.589·46-s − 0.875·47-s + 0.565·50-s + 1.23·53-s − 0.674·55-s + 0.656·58-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.00207402675856, −12.41559120508264, −12.06169094434786, −11.65858126995980, −11.16344383941827, −10.84354277588685, −10.23487637726977, −9.730572511260658, −9.378528063382449, −8.839697758653184, −8.420795438246366, −8.030559815251994, −7.462024201948886, −7.005499395762618, −6.402764375213438, −6.107573575477465, −5.690317330194286, −4.653423654894491, −4.323979979553031, −3.807908587272419, −3.369895615693776, −2.502267496764034, −1.939150104868605, −1.517094087181213, −0.6612752848281907, 0,
0.6612752848281907, 1.517094087181213, 1.939150104868605, 2.502267496764034, 3.369895615693776, 3.807908587272419, 4.323979979553031, 4.653423654894491, 5.690317330194286, 6.107573575477465, 6.402764375213438, 7.005499395762618, 7.462024201948886, 8.030559815251994, 8.420795438246366, 8.839697758653184, 9.378528063382449, 9.730572511260658, 10.23487637726977, 10.84354277588685, 11.16344383941827, 11.65858126995980, 12.06169094434786, 12.41559120508264, 13.00207402675856
