| L(s) = 1 | + 2-s − 2.80·3-s + 4-s + 0.676·5-s − 2.80·6-s + 3.76·7-s + 8-s + 4.84·9-s + 0.676·10-s − 0.631·11-s − 2.80·12-s − 6.22·13-s + 3.76·14-s − 1.89·15-s + 16-s + 3.39·17-s + 4.84·18-s + 1.27·19-s + 0.676·20-s − 10.5·21-s − 0.631·22-s + 3.84·23-s − 2.80·24-s − 4.54·25-s − 6.22·26-s − 5.17·27-s + 3.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.302·5-s − 1.14·6-s + 1.42·7-s + 0.353·8-s + 1.61·9-s + 0.213·10-s − 0.190·11-s − 0.808·12-s − 1.72·13-s + 1.00·14-s − 0.489·15-s + 0.250·16-s + 0.823·17-s + 1.14·18-s + 0.292·19-s + 0.151·20-s − 2.29·21-s − 0.134·22-s + 0.801·23-s − 0.571·24-s − 0.908·25-s − 1.22·26-s − 0.995·27-s + 0.710·28-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 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 0.676T + 5T^{2} \) |
| 7 | \( 1 - 3.76T + 7T^{2} \) |
| 11 | \( 1 + 0.631T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 3.84T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 + 9.06T + 37T^{2} \) |
| 41 | \( 1 - 0.532T + 41T^{2} \) |
| 43 | \( 1 + 7.96T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 - 1.69T + 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 2.77T + 97T^{2} \) |
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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
−7.32428660574013478374814706115, −6.73554935307022271403313471308, −5.75548497303803185926307740621, −5.24763296030499904180796316016, −5.00602027990393512674281059653, −4.33487152653947840374074711157, −3.20061887354885536779764590586, −2.02798745564893811754849979594, −1.35965458896112650527834945098, 0,
1.35965458896112650527834945098, 2.02798745564893811754849979594, 3.20061887354885536779764590586, 4.33487152653947840374074711157, 5.00602027990393512674281059653, 5.24763296030499904180796316016, 5.75548497303803185926307740621, 6.73554935307022271403313471308, 7.32428660574013478374814706115
