| L(s) = 1 | − 1.34·2-s + 2.37·3-s − 0.190·4-s − 0.880·5-s − 3.19·6-s + 0.181·7-s + 2.94·8-s + 2.62·9-s + 1.18·10-s − 3.21·11-s − 0.450·12-s + 4.11·13-s − 0.243·14-s − 2.08·15-s − 3.58·16-s − 1.37·17-s − 3.53·18-s − 7.81·19-s + 0.167·20-s + 0.430·21-s + 4.32·22-s + 1.87·23-s + 6.98·24-s − 4.22·25-s − 5.54·26-s − 0.883·27-s − 0.0344·28-s + ⋯ |
| L(s) = 1 | − 0.951·2-s + 1.36·3-s − 0.0950·4-s − 0.393·5-s − 1.30·6-s + 0.0685·7-s + 1.04·8-s + 0.875·9-s + 0.374·10-s − 0.968·11-s − 0.130·12-s + 1.14·13-s − 0.0652·14-s − 0.539·15-s − 0.895·16-s − 0.333·17-s − 0.833·18-s − 1.79·19-s + 0.0374·20-s + 0.0938·21-s + 0.921·22-s + 0.391·23-s + 1.42·24-s − 0.844·25-s − 1.08·26-s − 0.170·27-s − 0.00651·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2011 | \( 1 + T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 + 0.880T + 5T^{2} \) |
| 7 | \( 1 - 0.181T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 - 1.99T + 59T^{2} \) |
| 61 | \( 1 - 9.10T + 61T^{2} \) |
| 67 | \( 1 - 8.59T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 2.06T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 2.19T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.685644607607994900738102792507, −8.087176992514871112983349121717, −7.79679659935971867350227269983, −6.76234087908890336288301788043, −5.58693618144016237964191562346, −4.30334982151721800284045174593, −3.78158850276225305338488311992, −2.56132063180840452678382347914, −1.67263860544362670780449792434, 0,
1.67263860544362670780449792434, 2.56132063180840452678382347914, 3.78158850276225305338488311992, 4.30334982151721800284045174593, 5.58693618144016237964191562346, 6.76234087908890336288301788043, 7.79679659935971867350227269983, 8.087176992514871112983349121717, 8.685644607607994900738102792507
