| L(s) = 1 | + 1.84·2-s + 1.40·4-s − 2.16·5-s − 7-s − 1.10·8-s − 3.99·10-s + 5.73·11-s + 4.52·13-s − 1.84·14-s − 4.83·16-s + 5.13·17-s − 5.56·19-s − 3.03·20-s + 10.5·22-s + 0.333·23-s − 0.307·25-s + 8.34·26-s − 1.40·28-s − 5.19·29-s + 7.31·31-s − 6.71·32-s + 9.47·34-s + 2.16·35-s + 0.521·37-s − 10.2·38-s + 2.39·40-s − 10.1·41-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.700·4-s − 0.968·5-s − 0.377·7-s − 0.390·8-s − 1.26·10-s + 1.72·11-s + 1.25·13-s − 0.492·14-s − 1.20·16-s + 1.24·17-s − 1.27·19-s − 0.678·20-s + 2.25·22-s + 0.0696·23-s − 0.0614·25-s + 1.63·26-s − 0.264·28-s − 0.964·29-s + 1.31·31-s − 1.18·32-s + 1.62·34-s + 0.366·35-s + 0.0857·37-s − 1.66·38-s + 0.378·40-s − 1.58·41-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
| \(L(1)\) |
\(\approx\) |
\(3.368093928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.368093928\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.84T + 2T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 - 0.333T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 - 0.521T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 - 3.25T + 47T^{2} \) |
| 53 | \( 1 + 3.96T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 - 6.52T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 0.863T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.76678948727401738843643710104, −6.79534283027545595228729871688, −6.30773128328396108798415328812, −5.85563159965999356446143802115, −4.81329696745669627860418677552, −4.12367412174800643689848925580, −3.61994496515808068075159370246, −3.29371266587811784515963118866, −1.93410604438404015748324504936, −0.75578958367105000140496072505,
0.75578958367105000140496072505, 1.93410604438404015748324504936, 3.29371266587811784515963118866, 3.61994496515808068075159370246, 4.12367412174800643689848925580, 4.81329696745669627860418677552, 5.85563159965999356446143802115, 6.30773128328396108798415328812, 6.79534283027545595228729871688, 7.76678948727401738843643710104
