| L(s) = 1 | − 1.56·2-s + 0.452·4-s − 2.71·5-s + 7-s + 2.42·8-s + 4.24·10-s + 3.86·11-s + 6.27·13-s − 1.56·14-s − 4.70·16-s + 6.52·17-s − 5.34·19-s − 1.22·20-s − 6.05·22-s + 1.10·23-s + 2.35·25-s − 9.82·26-s + 0.452·28-s + 1.30·29-s − 5.39·31-s + 2.51·32-s − 10.2·34-s − 2.71·35-s − 4.76·37-s + 8.37·38-s − 6.57·40-s − 9.81·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.226·4-s − 1.21·5-s + 0.377·7-s + 0.856·8-s + 1.34·10-s + 1.16·11-s + 1.73·13-s − 0.418·14-s − 1.17·16-s + 1.58·17-s − 1.22·19-s − 0.274·20-s − 1.29·22-s + 0.230·23-s + 0.471·25-s − 1.92·26-s + 0.0855·28-s + 0.242·29-s − 0.968·31-s + 0.444·32-s − 1.75·34-s − 0.458·35-s − 0.782·37-s + 1.35·38-s − 1.03·40-s − 1.53·41-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 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 + 2.71T + 5T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 6.27T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 - 1.10T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 + 5.39T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 + 9.81T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 6.54T + 83T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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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
−7.74270061141760790559844926179, −7.02374780194231207120716147342, −6.37217155721798465168016913859, −5.43148055873361587927199230543, −4.44562280233247032741988903451, −3.83367938103252383155472973810, −3.34716546389154704446866109455, −1.66219214752663762901340077563, −1.17999037487354170107827893960, 0,
1.17999037487354170107827893960, 1.66219214752663762901340077563, 3.34716546389154704446866109455, 3.83367938103252383155472973810, 4.44562280233247032741988903451, 5.43148055873361587927199230543, 6.37217155721798465168016913859, 7.02374780194231207120716147342, 7.74270061141760790559844926179
