| L(s) = 1 | + 2·2-s − 3·3-s + 2·4-s + 5-s − 6·6-s − 7-s + 6·9-s + 2·10-s − 6·12-s + 6·13-s − 2·14-s − 3·15-s − 4·16-s + 4·17-s + 12·18-s − 19-s + 2·20-s + 3·21-s + 9·23-s + 25-s + 12·26-s − 9·27-s − 2·28-s + 5·29-s − 6·30-s − 31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 4-s + 0.447·5-s − 2.44·6-s − 0.377·7-s + 2·9-s + 0.632·10-s − 1.73·12-s + 1.66·13-s − 0.534·14-s − 0.774·15-s − 16-s + 0.970·17-s + 2.82·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s + 1.87·23-s + 1/5·25-s + 2.35·26-s − 1.73·27-s − 0.377·28-s + 0.928·29-s − 1.09·30-s − 0.179·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009102876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009102876\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 - T \) | |
| 151 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−10.89769315994247068795702748681, −9.821599681595636350143055491402, −8.714570290388150728735372265800, −7.05582089620535366417885455704, −6.37645990073359166886674809389, −5.76023063565317861683426689927, −5.15005654462266149294399477060, −4.18842039426849904071575242795, −3.11257095712003080603672911220, −1.12908986678937476948930399684,
1.12908986678937476948930399684, 3.11257095712003080603672911220, 4.18842039426849904071575242795, 5.15005654462266149294399477060, 5.76023063565317861683426689927, 6.37645990073359166886674809389, 7.05582089620535366417885455704, 8.714570290388150728735372265800, 9.821599681595636350143055491402, 10.89769315994247068795702748681
