| L(s) = 1 | + 256·2-s − 4.20e3·3-s + 6.55e4·4-s − 3.90e5·5-s − 1.07e6·6-s + 1.23e7·7-s + 1.67e7·8-s − 1.11e8·9-s − 1.00e8·10-s + 1.21e9·11-s − 2.75e8·12-s + 2.42e9·13-s + 3.15e9·14-s + 1.64e9·15-s + 4.29e9·16-s − 1.73e10·17-s − 2.85e10·18-s + 1.15e11·19-s − 2.56e10·20-s − 5.17e10·21-s + 3.11e11·22-s + 5.47e11·23-s − 7.04e10·24-s + 1.52e11·25-s + 6.20e11·26-s + 1.01e12·27-s + 8.07e11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.369·3-s + 0.5·4-s − 0.447·5-s − 0.261·6-s + 0.807·7-s + 0.353·8-s − 0.863·9-s − 0.316·10-s + 1.70·11-s − 0.184·12-s + 0.824·13-s + 0.571·14-s + 0.165·15-s + 0.250·16-s − 0.603·17-s − 0.610·18-s + 1.55·19-s − 0.223·20-s − 0.298·21-s + 1.20·22-s + 1.45·23-s − 0.130·24-s + 0.200·25-s + 0.582·26-s + 0.688·27-s + 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.786464087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.786464087\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 256T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 3 | \( 1 + 4.20e3T + 1.29e8T^{2} \) |
| 7 | \( 1 - 1.23e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.21e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.42e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.73e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.15e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.47e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 2.02e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.37e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.48e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.37e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 2.49e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.57e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 5.06e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.29e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 4.35e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.50e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.06e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 4.56e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.40e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.52e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 6.68e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 6.24e16T + 5.95e33T^{2} \) |
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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
−16.49452189855450298359084947718, −14.91810322526225841113093940138, −13.82863554692487995426831217217, −11.85444941196240143799885967530, −11.21093525111772113575974931888, −8.716239185577053457074559797551, −6.76098960845187442525068610138, −5.11636671786388053697245566920, −3.48709245950675540928065591627, −1.22162989460489883800798337344,
1.22162989460489883800798337344, 3.48709245950675540928065591627, 5.11636671786388053697245566920, 6.76098960845187442525068610138, 8.716239185577053457074559797551, 11.21093525111772113575974931888, 11.85444941196240143799885967530, 13.82863554692487995426831217217, 14.91810322526225841113093940138, 16.49452189855450298359084947718
