| L(s) = 1 | − 2-s − 2.62·3-s + 4-s + 0.854·5-s + 2.62·6-s + 1.79·7-s − 8-s + 3.89·9-s − 0.854·10-s + 1.39·11-s − 2.62·12-s + 4.58·13-s − 1.79·14-s − 2.24·15-s + 16-s + 4.16·17-s − 3.89·18-s + 7.03·19-s + 0.854·20-s − 4.70·21-s − 1.39·22-s − 6.43·23-s + 2.62·24-s − 4.26·25-s − 4.58·26-s − 2.36·27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.51·3-s + 0.5·4-s + 0.382·5-s + 1.07·6-s + 0.677·7-s − 0.353·8-s + 1.29·9-s − 0.270·10-s + 0.421·11-s − 0.758·12-s + 1.27·13-s − 0.479·14-s − 0.579·15-s + 0.250·16-s + 1.00·17-s − 0.919·18-s + 1.61·19-s + 0.191·20-s − 1.02·21-s − 0.298·22-s − 1.34·23-s + 0.536·24-s − 0.853·25-s − 0.900·26-s − 0.454·27-s + 0.338·28-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
| \(L(1)\) |
\(\approx\) |
\(1.367590601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367590601\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 - 0.854T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 - 7.42T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 + 5.20T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83679814138151172407773232373, −7.13231713351672777220006307835, −6.28857784638094024097006414100, −5.75656454112913074015325092969, −5.46899675456719519647008023724, −4.37946658233223830177954450037, −3.64039718448894467619184220234, −2.37284633198080235052114067757, −1.19076034844882960372763803059, −0.890305286771133774095764479265,
0.890305286771133774095764479265, 1.19076034844882960372763803059, 2.37284633198080235052114067757, 3.64039718448894467619184220234, 4.37946658233223830177954450037, 5.46899675456719519647008023724, 5.75656454112913074015325092969, 6.28857784638094024097006414100, 7.13231713351672777220006307835, 7.83679814138151172407773232373
