| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 7-s + 8-s + 9-s − 3·10-s + 11-s + 12-s + 13-s − 14-s − 3·15-s + 16-s + 4·17-s + 18-s + 19-s − 3·20-s − 21-s + 22-s + 24-s + 4·25-s + 26-s + 27-s − 28-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.229·19-s − 0.670·20-s − 0.218·21-s + 0.213·22-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.69260010016461, −14.31962635420687, −13.50925493684890, −13.22880187284354, −12.66405641071413, −12.04870532820695, −11.72529557172215, −11.35004329361566, −10.62034000775054, −9.963054950079429, −9.669406486298889, −8.741610010030567, −8.256831669829274, −7.979970089918204, −7.118266737454077, −6.969565690239665, −6.218128420693969, −5.472180884834394, −4.897702193515929, −4.250151014566061, −3.574736490019379, −3.437874122565307, −2.762972527940284, −1.811702344639149, −1.051070306651808, 0,
1.051070306651808, 1.811702344639149, 2.762972527940284, 3.437874122565307, 3.574736490019379, 4.250151014566061, 4.897702193515929, 5.472180884834394, 6.218128420693969, 6.969565690239665, 7.118266737454077, 7.979970089918204, 8.256831669829274, 8.741610010030567, 9.669406486298889, 9.963054950079429, 10.62034000775054, 11.35004329361566, 11.72529557172215, 12.04870532820695, 12.66405641071413, 13.22880187284354, 13.50925493684890, 14.31962635420687, 14.69260010016461
