| L(s) = 1 | − 12.6·2-s + 75.4·3-s + 31.1·4-s − 68.7·5-s − 952.·6-s + 325.·7-s + 1.22e3·8-s + 3.51e3·9-s + 867.·10-s + 4.36e3·11-s + 2.35e3·12-s + 1.04e4·13-s − 4.11e3·14-s − 5.18e3·15-s − 1.94e4·16-s − 3.33e3·17-s − 4.42e4·18-s − 6.85e3·19-s − 2.14e3·20-s + 2.45e4·21-s − 5.51e4·22-s + 1.00e5·23-s + 9.21e4·24-s − 7.34e4·25-s − 1.31e5·26-s + 9.98e4·27-s + 1.01e4·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 1.61·3-s + 0.243·4-s − 0.245·5-s − 1.79·6-s + 0.359·7-s + 0.843·8-s + 1.60·9-s + 0.274·10-s + 0.989·11-s + 0.393·12-s + 1.31·13-s − 0.400·14-s − 0.396·15-s − 1.18·16-s − 0.164·17-s − 1.78·18-s − 0.229·19-s − 0.0599·20-s + 0.579·21-s − 1.10·22-s + 1.72·23-s + 1.36·24-s − 0.939·25-s − 1.46·26-s + 0.976·27-s + 0.0875·28-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
| \(L(4)\) |
\(\approx\) |
\(1.508092927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508092927\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 12.6T + 128T^{2} \) |
| 3 | \( 1 - 75.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 68.7T + 7.81e4T^{2} \) |
| 7 | \( 1 - 325.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.36e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.04e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.33e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 1.00e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.10e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.96e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.55e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.34e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.39e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.68e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.23e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.21e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.18e6T + 8.07e13T^{2} \) |
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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
−17.08882379722442652365313747228, −15.57946195261985804688661253065, −14.32121460094942750280753187069, −13.25648304156532683972156994701, −10.97864308378633394663366194135, −9.239342579002795402779916515612, −8.650622577237020985812179992136, −7.36864248917683609476784569896, −3.83080799470305582793087873246, −1.54202250694140841979501450718,
1.54202250694140841979501450718, 3.83080799470305582793087873246, 7.36864248917683609476784569896, 8.650622577237020985812179992136, 9.239342579002795402779916515612, 10.97864308378633394663366194135, 13.25648304156532683972156994701, 14.32121460094942750280753187069, 15.57946195261985804688661253065, 17.08882379722442652365313747228
