| L(s) = 1 | − 2-s + 0.188·3-s + 4-s + 2.47·5-s − 0.188·6-s − 1.95·7-s − 8-s − 2.96·9-s − 2.47·10-s + 3.31·11-s + 0.188·12-s + 4.02·13-s + 1.95·14-s + 0.466·15-s + 16-s − 5.28·17-s + 2.96·18-s + 19-s + 2.47·20-s − 0.369·21-s − 3.31·22-s − 6.28·23-s − 0.188·24-s + 1.10·25-s − 4.02·26-s − 1.12·27-s − 1.95·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.108·3-s + 0.5·4-s + 1.10·5-s − 0.0770·6-s − 0.739·7-s − 0.353·8-s − 0.988·9-s − 0.781·10-s + 0.998·11-s + 0.0544·12-s + 1.11·13-s + 0.522·14-s + 0.120·15-s + 0.250·16-s − 1.28·17-s + 0.698·18-s + 0.229·19-s + 0.552·20-s − 0.0805·21-s − 0.706·22-s − 1.31·23-s − 0.0385·24-s + 0.220·25-s − 0.790·26-s − 0.216·27-s − 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.425487435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425487435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.188T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 + 9.51T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 59 | \( 1 + 5.95T + 59T^{2} \) |
| 61 | \( 1 - 7.52T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 9.58T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.908569922089172241132003717898, −7.11128850189351112580082316039, −6.28170277575606561064089699082, −6.01854399461929983310257219465, −5.43326095499085253549113409299, −3.96469793788824168971247068622, −3.53225187697117179940592969406, −2.28962457476569405505764991065, −1.92882950617196797292437439069, −0.63308169807341668209653316495,
0.63308169807341668209653316495, 1.92882950617196797292437439069, 2.28962457476569405505764991065, 3.53225187697117179940592969406, 3.96469793788824168971247068622, 5.43326095499085253549113409299, 6.01854399461929983310257219465, 6.28170277575606561064089699082, 7.11128850189351112580082316039, 7.908569922089172241132003717898
