| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 5·7-s − 8-s − 2·9-s − 4·11-s − 12-s + 5·14-s + 16-s − 17-s + 2·18-s + 2·19-s + 5·21-s + 4·22-s + 8·23-s + 24-s + 5·27-s − 5·28-s + 5·31-s − 32-s + 4·33-s + 34-s − 2·36-s − 12·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s − 0.353·8-s − 2/3·9-s − 1.20·11-s − 0.288·12-s + 1.33·14-s + 1/4·16-s − 0.242·17-s + 0.471·18-s + 0.458·19-s + 1.09·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.962·27-s − 0.944·28-s + 0.898·31-s − 0.176·32-s + 0.696·33-s + 0.171·34-s − 1/3·36-s − 1.97·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.39878471152515, −13.06642298953312, −12.64381070117720, −12.21764150265281, −11.63759936665303, −11.14468492544767, −10.56801118499685, −10.36822209629433, −9.814896924279251, −9.246640714088266, −8.907033807099115, −8.410210161309601, −7.724331025321174, −7.129239848578275, −6.807573330770868, −6.300131603392503, −5.698821095485672, −5.384496527212867, −4.760801180900756, −3.856102029112935, −3.212955307226022, −2.708191946864584, −2.526052656776540, −1.253885348567076, −0.5548094912255499, 0,
0.5548094912255499, 1.253885348567076, 2.526052656776540, 2.708191946864584, 3.212955307226022, 3.856102029112935, 4.760801180900756, 5.384496527212867, 5.698821095485672, 6.300131603392503, 6.807573330770868, 7.129239848578275, 7.724331025321174, 8.410210161309601, 8.907033807099115, 9.246640714088266, 9.814896924279251, 10.36822209629433, 10.56801118499685, 11.14468492544767, 11.63759936665303, 12.21764150265281, 12.64381070117720, 13.06642298953312, 13.39878471152515
