| L(s) = 1 | + 1.94·2-s + 1.78·4-s + 1.34·5-s + 7-s − 0.426·8-s + 2.61·10-s − 0.671·11-s − 2.12·13-s + 1.94·14-s − 4.39·16-s + 1.91·17-s − 4.48·19-s + 2.39·20-s − 1.30·22-s − 4.40·23-s − 3.19·25-s − 4.12·26-s + 1.78·28-s − 2.36·29-s + 1.43·31-s − 7.68·32-s + 3.71·34-s + 1.34·35-s + 5.60·37-s − 8.71·38-s − 0.572·40-s − 4.45·41-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.890·4-s + 0.600·5-s + 0.377·7-s − 0.150·8-s + 0.825·10-s − 0.202·11-s − 0.589·13-s + 0.519·14-s − 1.09·16-s + 0.463·17-s − 1.02·19-s + 0.534·20-s − 0.278·22-s − 0.918·23-s − 0.639·25-s − 0.809·26-s + 0.336·28-s − 0.438·29-s + 0.258·31-s − 1.35·32-s + 0.637·34-s + 0.226·35-s + 0.921·37-s − 1.41·38-s − 0.0905·40-s − 0.696·41-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 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 + 0.671T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 + 0.516T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 + 8.45T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.29127202262738777774028677924, −6.47862223307226735798124731683, −5.95446596988286599591077732891, −5.32781857748533181947599331294, −4.69335832422527238468426988348, −4.05198500879192561854055878491, −3.24337216986353082071501796890, −2.36593879062252331537288857720, −1.73371554948533187685025065582, 0,
1.73371554948533187685025065582, 2.36593879062252331537288857720, 3.24337216986353082071501796890, 4.05198500879192561854055878491, 4.69335832422527238468426988348, 5.32781857748533181947599331294, 5.95446596988286599591077732891, 6.47862223307226735798124731683, 7.29127202262738777774028677924
