| L(s) = 1 | − 4.37·2-s − 63.9·3-s − 108.·4-s + 301.·5-s + 279.·6-s + 615.·7-s + 1.03e3·8-s + 1.90e3·9-s − 1.32e3·10-s − 3.51e3·11-s + 6.96e3·12-s + 1.36e4·13-s − 2.69e3·14-s − 1.93e4·15-s + 9.40e3·16-s + 1.62e3·17-s − 8.33e3·18-s − 6.85e3·19-s − 3.28e4·20-s − 3.93e4·21-s + 1.53e4·22-s + 3.99e4·23-s − 6.62e4·24-s + 1.30e4·25-s − 5.97e4·26-s + 1.79e4·27-s − 6.69e4·28-s + ⋯ |
| L(s) = 1 | − 0.386·2-s − 1.36·3-s − 0.850·4-s + 1.08·5-s + 0.529·6-s + 0.677·7-s + 0.715·8-s + 0.871·9-s − 0.417·10-s − 0.796·11-s + 1.16·12-s + 1.72·13-s − 0.262·14-s − 1.47·15-s + 0.573·16-s + 0.0802·17-s − 0.337·18-s − 0.229·19-s − 0.918·20-s − 0.927·21-s + 0.308·22-s + 0.684·23-s − 0.978·24-s + 0.167·25-s − 0.666·26-s + 0.175·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8664019883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8664019883\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + 6.85e3T \) |
| good | 2 | \( 1 + 4.37T + 128T^{2} \) |
| 3 | \( 1 + 63.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 301.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 615.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.51e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.36e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.62e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 3.99e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.86e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.80e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.11e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.31e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.23e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.89e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.29e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 8.07e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.93e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.25e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.08e6T + 8.07e13T^{2} \) |
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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
−17.41187835389935736082385811206, −16.05430795352143081121041706758, −13.98826074065809079606278733942, −12.94824962031830371772827301470, −11.13722553895269912612677562858, −10.14276571792529876243684232065, −8.453238118468240933276458669874, −6.09251106917340881319653750342, −4.90790053446088621669804076203, −1.04250921838190985700160682832,
1.04250921838190985700160682832, 4.90790053446088621669804076203, 6.09251106917340881319653750342, 8.453238118468240933276458669874, 10.14276571792529876243684232065, 11.13722553895269912612677562858, 12.94824962031830371772827301470, 13.98826074065809079606278733942, 16.05430795352143081121041706758, 17.41187835389935736082385811206
