| L(s) = 1 | + 2-s − 2.73·3-s + 4-s + 1.40·5-s − 2.73·6-s + 2.82·7-s + 8-s + 4.47·9-s + 1.40·10-s + 1.99·11-s − 2.73·12-s − 3.41·13-s + 2.82·14-s − 3.82·15-s + 16-s − 3.91·17-s + 4.47·18-s + 7.80·19-s + 1.40·20-s − 7.71·21-s + 1.99·22-s − 4.09·23-s − 2.73·24-s − 3.03·25-s − 3.41·26-s − 4.03·27-s + 2.82·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.57·3-s + 0.5·4-s + 0.626·5-s − 1.11·6-s + 1.06·7-s + 0.353·8-s + 1.49·9-s + 0.442·10-s + 0.600·11-s − 0.789·12-s − 0.947·13-s + 0.754·14-s − 0.988·15-s + 0.250·16-s − 0.948·17-s + 1.05·18-s + 1.78·19-s + 0.313·20-s − 1.68·21-s + 0.424·22-s − 0.853·23-s − 0.558·24-s − 0.607·25-s − 0.670·26-s − 0.775·27-s + 0.533·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 4003 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 1.99T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 - 7.80T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 + 6.29T + 31T^{2} \) |
| 37 | \( 1 - 1.93T + 37T^{2} \) |
| 41 | \( 1 + 6.35T + 41T^{2} \) |
| 43 | \( 1 + 0.471T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 9.61T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 1.62T + 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.54T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 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
−7.47280074109035679924955215560, −6.51084392171721039311942361529, −5.89596891795611391047858505138, −5.44858228790255633710190270302, −4.77135211339499852787893132430, −4.35509945742666878608069440224, −3.23038334505435454049331493095, −1.93150064894812767601619000096, −1.46330629871073875050701492495, 0,
1.46330629871073875050701492495, 1.93150064894812767601619000096, 3.23038334505435454049331493095, 4.35509945742666878608069440224, 4.77135211339499852787893132430, 5.44858228790255633710190270302, 5.89596891795611391047858505138, 6.51084392171721039311942361529, 7.47280074109035679924955215560
