| L(s) = 1 | − 11.7·2-s + 43.4·3-s + 9.96·4-s − 389.·5-s − 510.·6-s − 1.24e3·7-s + 1.38e3·8-s − 296.·9-s + 4.57e3·10-s + 1.33e3·11-s + 433.·12-s − 3.84e3·13-s + 1.46e4·14-s − 1.69e4·15-s − 1.75e4·16-s + 2.41e4·17-s + 3.48e3·18-s + 5.45e3·19-s − 3.88e3·20-s − 5.43e4·21-s − 1.56e4·22-s − 6.39e4·23-s + 6.02e4·24-s + 7.39e4·25-s + 4.51e4·26-s − 1.07e5·27-s − 1.24e4·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.929·3-s + 0.0778·4-s − 1.39·5-s − 0.965·6-s − 1.37·7-s + 0.957·8-s − 0.135·9-s + 1.44·10-s + 0.301·11-s + 0.0723·12-s − 0.484·13-s + 1.42·14-s − 1.29·15-s − 1.07·16-s + 1.19·17-s + 0.140·18-s + 0.182·19-s − 0.108·20-s − 1.27·21-s − 0.313·22-s − 1.09·23-s + 0.890·24-s + 0.946·25-s + 0.503·26-s − 1.05·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 + 11.7T + 128T^{2} \) |
| 3 | \( 1 - 43.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 389.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.24e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 3.84e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.41e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.45e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.89e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.78e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.14e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.38e6T + 8.07e13T^{2} \) |
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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
−18.74262985339869049922574548668, −16.78325590181124904257958239418, −15.67130304666426526268675678106, −14.02148719610647872437713388052, −12.17406919998667925722274616736, −10.02117074911111807773560520951, −8.684188947214052193755336815812, −7.47090187234197490854674575306, −3.53619294295583610418783673825, 0,
3.53619294295583610418783673825, 7.47090187234197490854674575306, 8.684188947214052193755336815812, 10.02117074911111807773560520951, 12.17406919998667925722274616736, 14.02148719610647872437713388052, 15.67130304666426526268675678106, 16.78325590181124904257958239418, 18.74262985339869049922574548668
