| L(s) = 1 | + 7.78·2-s + 37.1·3-s − 67.3·4-s + 383.·5-s + 289.·6-s + 1.02e3·7-s − 1.52e3·8-s − 804.·9-s + 2.98e3·10-s + 7.46e3·11-s − 2.50e3·12-s − 5.46e3·13-s + 7.98e3·14-s + 1.42e4·15-s − 3.22e3·16-s − 3.32e4·17-s − 6.26e3·18-s − 6.85e3·19-s − 2.58e4·20-s + 3.81e4·21-s + 5.81e4·22-s − 8.74e4·23-s − 5.65e4·24-s + 6.90e4·25-s − 4.25e4·26-s − 1.11e5·27-s − 6.90e4·28-s + ⋯ |
| L(s) = 1 | + 0.688·2-s + 0.795·3-s − 0.526·4-s + 1.37·5-s + 0.547·6-s + 1.12·7-s − 1.05·8-s − 0.367·9-s + 0.944·10-s + 1.69·11-s − 0.418·12-s − 0.690·13-s + 0.777·14-s + 1.09·15-s − 0.196·16-s − 1.64·17-s − 0.253·18-s − 0.229·19-s − 0.722·20-s + 0.898·21-s + 1.16·22-s − 1.49·23-s − 0.835·24-s + 0.884·25-s − 0.475·26-s − 1.08·27-s − 0.594·28-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
| \(L(4)\) |
\(\approx\) |
\(2.895307206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.895307206\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 7.78T + 128T^{2} \) |
| 3 | \( 1 - 37.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 383.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.02e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.46e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.46e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.32e4T + 4.10e8T^{2} \) |
| 23 | \( 1 + 8.74e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.20e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 6.02e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.37e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.96e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.37e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.98e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.28e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.06e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.44e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.66e6T + 8.07e13T^{2} \) |
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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
−17.37034170461570256329497769307, −14.78803870963300729188436900504, −14.22512355627036296678221205535, −13.45135220002479023894052899581, −11.72172593109527982606534706144, −9.555825404712745183203054711338, −8.607078797324282986253531320855, −6.08569611283034673622064074388, −4.37643857771174733650272688185, −2.12306934123709445141980371028,
2.12306934123709445141980371028, 4.37643857771174733650272688185, 6.08569611283034673622064074388, 8.607078797324282986253531320855, 9.555825404712745183203054711338, 11.72172593109527982606534706144, 13.45135220002479023894052899581, 14.22512355627036296678221205535, 14.78803870963300729188436900504, 17.37034170461570256329497769307
