| L(s) = 1 | + 2-s − 2.01·3-s + 4-s + 2.07·5-s − 2.01·6-s − 1.64·7-s + 8-s + 1.05·9-s + 2.07·10-s − 2.89·11-s − 2.01·12-s − 2.54·13-s − 1.64·14-s − 4.18·15-s + 16-s + 1.07·17-s + 1.05·18-s + 6.85·19-s + 2.07·20-s + 3.30·21-s − 2.89·22-s − 1.82·23-s − 2.01·24-s − 0.679·25-s − 2.54·26-s + 3.91·27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.16·3-s + 0.5·4-s + 0.929·5-s − 0.822·6-s − 0.620·7-s + 0.353·8-s + 0.353·9-s + 0.657·10-s − 0.874·11-s − 0.581·12-s − 0.706·13-s − 0.438·14-s − 1.08·15-s + 0.250·16-s + 0.261·17-s + 0.249·18-s + 1.57·19-s + 0.464·20-s + 0.721·21-s − 0.618·22-s − 0.381·23-s − 0.411·24-s − 0.135·25-s − 0.499·26-s + 0.752·27-s − 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.01T + 3T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 3.20T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 0.216T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 + 1.16T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 1.87T + 79T^{2} \) |
| 83 | \( 1 + 2.72T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.35996335023393638539579747244, −6.51986647871792855547607602760, −5.78472513650286721456463791146, −5.57359732137245268298205883233, −4.98974774011187278301796744640, −4.06490268059707297122691011627, −3.00042514203950626765806673466, −2.43062457448402946207330436275, −1.24522828273982371968776379309, 0,
1.24522828273982371968776379309, 2.43062457448402946207330436275, 3.00042514203950626765806673466, 4.06490268059707297122691011627, 4.98974774011187278301796744640, 5.57359732137245268298205883233, 5.78472513650286721456463791146, 6.51986647871792855547607602760, 7.35996335023393638539579747244
