| L(s) = 1 | − 0.691·2-s − 1.52·4-s − 0.236·5-s + 7-s + 2.43·8-s + 0.163·10-s − 4.73·11-s − 1.51·13-s − 0.691·14-s + 1.35·16-s + 4.88·17-s + 2.13·19-s + 0.360·20-s + 3.27·22-s + 1.62·23-s − 4.94·25-s + 1.05·26-s − 1.52·28-s − 0.481·29-s − 9.10·31-s − 5.81·32-s − 3.37·34-s − 0.236·35-s + 10.9·37-s − 1.47·38-s − 0.576·40-s − 4.78·41-s + ⋯ |
| L(s) = 1 | − 0.489·2-s − 0.760·4-s − 0.105·5-s + 0.377·7-s + 0.861·8-s + 0.0517·10-s − 1.42·11-s − 0.421·13-s − 0.184·14-s + 0.339·16-s + 1.18·17-s + 0.490·19-s + 0.0805·20-s + 0.697·22-s + 0.337·23-s − 0.988·25-s + 0.206·26-s − 0.287·28-s − 0.0894·29-s − 1.63·31-s − 1.02·32-s − 0.579·34-s − 0.0400·35-s + 1.80·37-s − 0.239·38-s − 0.0911·40-s − 0.746·41-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 | \( 1 + 0.691T + 2T^{2} \) |
| 5 | \( 1 + 0.236T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 + 0.481T + 29T^{2} \) |
| 31 | \( 1 + 9.10T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 9.66T + 97T^{2} \) |
| 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
−7.56329215535314457841628051932, −7.29581335746637612987421282161, −5.76150841196712808944009957956, −5.47718613775717386504204046568, −4.73213467374506407780500889808, −3.95956078633368946842248590821, −3.09909966367938572002132810958, −2.13217468065781512912438098371, −1.04873500308700696088563453412, 0,
1.04873500308700696088563453412, 2.13217468065781512912438098371, 3.09909966367938572002132810958, 3.95956078633368946842248590821, 4.73213467374506407780500889808, 5.47718613775717386504204046568, 5.76150841196712808944009957956, 7.29581335746637612987421282161, 7.56329215535314457841628051932
