| L(s) = 1 | + 1.33e16·2-s + 4.78e25·3-s + 1.72e31·4-s − 4.75e37·5-s + 6.40e41·6-s − 7.19e44·7-s − 1.94e48·8-s + 1.15e51·9-s − 6.36e53·10-s + 5.31e55·11-s + 8.23e56·12-s + 6.03e59·13-s − 9.63e60·14-s − 2.27e63·15-s − 2.88e64·16-s + 1.95e65·17-s + 1.55e67·18-s + 1.51e67·19-s − 8.18e68·20-s − 3.43e70·21-s + 7.12e71·22-s + 7.78e72·23-s − 9.28e73·24-s + 1.64e75·25-s + 8.08e75·26-s + 1.47e75·27-s − 1.23e76·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 1.42·3-s + 0.106·4-s − 1.91·5-s + 1.49·6-s − 0.440·7-s − 0.940·8-s + 1.02·9-s − 2.01·10-s + 1.02·11-s + 0.151·12-s + 1.53·13-s − 0.463·14-s − 2.72·15-s − 1.09·16-s + 0.289·17-s + 1.08·18-s + 0.0583·19-s − 0.203·20-s − 0.627·21-s + 1.07·22-s + 1.09·23-s − 1.33·24-s + 2.66·25-s + 1.60·26-s + 0.0390·27-s − 0.0467·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(108-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(54)\) |
\(\approx\) |
\(4.086160506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.086160506\) |
| \(L(\frac{109}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.33e16T + 1.62e32T^{2} \) |
| 3 | \( 1 - 4.78e25T + 1.12e51T^{2} \) |
| 5 | \( 1 + 4.75e37T + 6.16e74T^{2} \) |
| 7 | \( 1 + 7.19e44T + 2.66e90T^{2} \) |
| 11 | \( 1 - 5.31e55T + 2.68e111T^{2} \) |
| 13 | \( 1 - 6.03e59T + 1.55e119T^{2} \) |
| 17 | \( 1 - 1.95e65T + 4.55e131T^{2} \) |
| 19 | \( 1 - 1.51e67T + 6.70e136T^{2} \) |
| 23 | \( 1 - 7.78e72T + 5.06e145T^{2} \) |
| 29 | \( 1 - 2.54e78T + 2.99e156T^{2} \) |
| 31 | \( 1 - 1.31e79T + 3.76e159T^{2} \) |
| 37 | \( 1 - 5.58e82T + 6.27e167T^{2} \) |
| 41 | \( 1 - 1.85e85T + 3.69e172T^{2} \) |
| 43 | \( 1 + 2.08e86T + 6.04e174T^{2} \) |
| 47 | \( 1 - 2.95e87T + 8.21e178T^{2} \) |
| 53 | \( 1 + 9.11e91T + 3.14e184T^{2} \) |
| 59 | \( 1 + 4.40e94T + 3.02e189T^{2} \) |
| 61 | \( 1 - 3.11e95T + 1.07e191T^{2} \) |
| 67 | \( 1 - 3.76e97T + 2.45e195T^{2} \) |
| 71 | \( 1 + 3.36e98T + 1.21e198T^{2} \) |
| 73 | \( 1 - 2.58e99T + 2.37e199T^{2} \) |
| 79 | \( 1 - 3.43e100T + 1.11e203T^{2} \) |
| 83 | \( 1 - 7.77e102T + 2.19e205T^{2} \) |
| 89 | \( 1 + 3.50e104T + 3.84e208T^{2} \) |
| 97 | \( 1 - 1.80e105T + 3.84e212T^{2} \) |
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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
−13.99723462454766305376759943919, −12.70161254833180280227215380351, −11.45309822519313885296275889671, −9.001862041026084636454433065689, −8.207643400145716909443967200370, −6.65193867549914588132742863192, −4.44788866796141811752045139878, −3.55168723766922161405068228353, −3.13322748712685785694912478442, −0.863441791992397739346135241278,
0.863441791992397739346135241278, 3.13322748712685785694912478442, 3.55168723766922161405068228353, 4.44788866796141811752045139878, 6.65193867549914588132742863192, 8.207643400145716909443967200370, 9.001862041026084636454433065689, 11.45309822519313885296275889671, 12.70161254833180280227215380351, 13.99723462454766305376759943919
