| L(s) = 1 | − 2-s + 1.86·3-s + 4-s − 2.95·5-s − 1.86·6-s − 4.89·7-s − 8-s + 0.465·9-s + 2.95·10-s − 0.0809·11-s + 1.86·12-s − 1.13·13-s + 4.89·14-s − 5.49·15-s + 16-s − 3.96·17-s − 0.465·18-s − 5.98·19-s − 2.95·20-s − 9.11·21-s + 0.0809·22-s + 0.0756·23-s − 1.86·24-s + 3.70·25-s + 1.13·26-s − 4.71·27-s − 4.89·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.07·3-s + 0.5·4-s − 1.31·5-s − 0.759·6-s − 1.85·7-s − 0.353·8-s + 0.155·9-s + 0.933·10-s − 0.0244·11-s + 0.537·12-s − 0.315·13-s + 1.30·14-s − 1.41·15-s + 0.250·16-s − 0.962·17-s − 0.109·18-s − 1.37·19-s − 0.659·20-s − 1.98·21-s + 0.0172·22-s + 0.0157·23-s − 0.379·24-s + 0.741·25-s + 0.223·26-s − 0.908·27-s − 0.925·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3987630456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3987630456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 0.0809T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 5.98T + 19T^{2} \) |
| 23 | \( 1 - 0.0756T + 23T^{2} \) |
| 29 | \( 1 + 8.62T + 29T^{2} \) |
| 31 | \( 1 - 0.692T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 17.0T + 73T^{2} \) |
| 79 | \( 1 - 1.97T + 79T^{2} \) |
| 83 | \( 1 - 5.05T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 - 1.15T + 97T^{2} \) |
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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
−8.502694643924421553704100714381, −7.83223391351610607519683215381, −7.18244005656469169236428751444, −6.57054843348111323344719377372, −5.72973847402968170920814451721, −4.09228815494724145507905398825, −3.82252208764579142358186707008, −2.83034787019858253077467010908, −2.28050120323583214398181322287, −0.34873954071793075534096824556,
0.34873954071793075534096824556, 2.28050120323583214398181322287, 2.83034787019858253077467010908, 3.82252208764579142358186707008, 4.09228815494724145507905398825, 5.72973847402968170920814451721, 6.57054843348111323344719377372, 7.18244005656469169236428751444, 7.83223391351610607519683215381, 8.502694643924421553704100714381
