| L(s) = 1 | − 2.07·2-s + 2.30·4-s − 1.11·5-s − 7-s − 0.632·8-s + 2.30·10-s − 4.88·11-s − 0.190·13-s + 2.07·14-s − 3.29·16-s − 0.273·17-s + 1.55·19-s − 2.56·20-s + 10.1·22-s − 2.18·23-s − 3.76·25-s + 0.396·26-s − 2.30·28-s + 2.52·29-s + 0.702·31-s + 8.10·32-s + 0.566·34-s + 1.11·35-s + 8.99·37-s − 3.22·38-s + 0.703·40-s + 0.921·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.15·4-s − 0.497·5-s − 0.377·7-s − 0.223·8-s + 0.729·10-s − 1.47·11-s − 0.0529·13-s + 0.554·14-s − 0.824·16-s − 0.0662·17-s + 0.356·19-s − 0.573·20-s + 2.15·22-s − 0.456·23-s − 0.752·25-s + 0.0776·26-s − 0.435·28-s + 0.469·29-s + 0.126·31-s + 1.43·32-s + 0.0972·34-s + 0.187·35-s + 1.47·37-s − 0.523·38-s + 0.111·40-s + 0.143·41-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 | \( 1 + 2.07T + 2T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 + 0.190T + 13T^{2} \) |
| 17 | \( 1 + 0.273T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 0.702T + 31T^{2} \) |
| 37 | \( 1 - 8.99T + 37T^{2} \) |
| 41 | \( 1 - 0.921T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 - 7.57T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 - 5.13T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.64332795880337360175621298068, −7.20050609738842357756456293650, −6.33077537099736548615412469812, −5.51885710625976077382128772716, −4.67613328253356244116826918570, −3.81750771244497716713875075967, −2.76428368003920756238672974499, −2.13273331981789788086963279573, −0.869675197856163341012637053505, 0,
0.869675197856163341012637053505, 2.13273331981789788086963279573, 2.76428368003920756238672974499, 3.81750771244497716713875075967, 4.67613328253356244116826918570, 5.51885710625976077382128772716, 6.33077537099736548615412469812, 7.20050609738842357756456293650, 7.64332795880337360175621298068
