| L(s) = 1 | + 2.15·2-s + 2.64·4-s + 0.239·5-s + 7-s + 1.40·8-s + 0.515·10-s + 3.99·11-s − 1.61·13-s + 2.15·14-s − 2.27·16-s − 4.61·17-s − 4.17·19-s + 0.633·20-s + 8.62·22-s − 5.22·23-s − 4.94·25-s − 3.48·26-s + 2.64·28-s − 9.10·29-s − 6.57·31-s − 7.71·32-s − 9.94·34-s + 0.239·35-s − 0.353·37-s − 8.99·38-s + 0.334·40-s + 7.72·41-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.32·4-s + 0.106·5-s + 0.377·7-s + 0.495·8-s + 0.163·10-s + 1.20·11-s − 0.448·13-s + 0.576·14-s − 0.569·16-s − 1.11·17-s − 0.957·19-s + 0.141·20-s + 1.83·22-s − 1.08·23-s − 0.988·25-s − 0.683·26-s + 0.500·28-s − 1.69·29-s − 1.18·31-s − 1.36·32-s − 1.70·34-s + 0.0404·35-s − 0.0581·37-s − 1.45·38-s + 0.0529·40-s + 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 5 | \( 1 - 0.239T + 5T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 0.353T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 - 0.655T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 7.05T + 61T^{2} \) |
| 67 | \( 1 + 0.167T + 67T^{2} \) |
| 71 | \( 1 + 6.98T + 71T^{2} \) |
| 73 | \( 1 + 2.79T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 6.75T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−7.22989843521190072365867196363, −6.51261173919209425637670934899, −6.02318747534033094262445361568, −5.34872772235442747434210834735, −4.51178375731342637049001376374, −4.00135965857059444806225029315, −3.50162832179824173452788866040, −2.15306947128086954361354234104, −1.91285762042880458483414421913, 0,
1.91285762042880458483414421913, 2.15306947128086954361354234104, 3.50162832179824173452788866040, 4.00135965857059444806225029315, 4.51178375731342637049001376374, 5.34872772235442747434210834735, 6.02318747534033094262445361568, 6.51261173919209425637670934899, 7.22989843521190072365867196363
