| L(s) = 1 | + 0.0561·2-s − 3-s − 1.99·4-s − 2.87·5-s − 0.0561·6-s + 3.53·7-s − 0.224·8-s + 9-s − 0.161·10-s − 11-s + 1.99·12-s + 1.66·13-s + 0.198·14-s + 2.87·15-s + 3.98·16-s − 6.51·17-s + 0.0561·18-s − 4.27·19-s + 5.73·20-s − 3.53·21-s − 0.0561·22-s − 8.02·23-s + 0.224·24-s + 3.23·25-s + 0.0934·26-s − 27-s − 7.06·28-s + ⋯ |
| L(s) = 1 | + 0.0397·2-s − 0.577·3-s − 0.998·4-s − 1.28·5-s − 0.0229·6-s + 1.33·7-s − 0.0793·8-s + 0.333·9-s − 0.0509·10-s − 0.301·11-s + 0.576·12-s + 0.461·13-s + 0.0531·14-s + 0.741·15-s + 0.995·16-s − 1.57·17-s + 0.0132·18-s − 0.979·19-s + 1.28·20-s − 0.772·21-s − 0.0119·22-s − 1.67·23-s + 0.0458·24-s + 0.647·25-s + 0.0183·26-s − 0.192·27-s − 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6679684639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6679684639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0561T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 + 8.02T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 0.826T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 8.11T + 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 - 2.54T + 73T^{2} \) |
| 79 | \( 1 + 6.81T + 79T^{2} \) |
| 83 | \( 1 - 0.562T + 83T^{2} \) |
| 89 | \( 1 - 7.23T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.756076260817029776608550304099, −8.377838655716976189886636270625, −7.87754642783983318006096789829, −6.82752264322639084250481401125, −5.86080869076096592629689686243, −4.77392332965399556269407395111, −4.41888326912056854809413181704, −3.75242993927728645544399452093, −2.06293456518184643711364593914, −0.55454303462644218755256018498,
0.55454303462644218755256018498, 2.06293456518184643711364593914, 3.75242993927728645544399452093, 4.41888326912056854809413181704, 4.77392332965399556269407395111, 5.86080869076096592629689686243, 6.82752264322639084250481401125, 7.87754642783983318006096789829, 8.377838655716976189886636270625, 8.756076260817029776608550304099
