| L(s) = 1 | + 0.203·2-s − 1.95·4-s + 2.16·5-s − 7-s − 0.804·8-s + 0.440·10-s − 2.03·11-s + 1.83·13-s − 0.203·14-s + 3.75·16-s + 2.26·17-s − 5.79·19-s − 4.24·20-s − 0.413·22-s − 2.00·23-s − 0.298·25-s + 0.373·26-s + 1.95·28-s + 2.70·29-s + 0.925·31-s + 2.37·32-s + 0.460·34-s − 2.16·35-s + 7.11·37-s − 1.17·38-s − 1.74·40-s + 3.83·41-s + ⋯ |
| L(s) = 1 | + 0.143·2-s − 0.979·4-s + 0.969·5-s − 0.377·7-s − 0.284·8-s + 0.139·10-s − 0.613·11-s + 0.509·13-s − 0.0543·14-s + 0.938·16-s + 0.549·17-s − 1.32·19-s − 0.949·20-s − 0.0882·22-s − 0.417·23-s − 0.0596·25-s + 0.0732·26-s + 0.370·28-s + 0.501·29-s + 0.166·31-s + 0.419·32-s + 0.0789·34-s − 0.366·35-s + 1.16·37-s − 0.191·38-s − 0.275·40-s + 0.598·41-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 | \( 1 - 0.203T + 2T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 + 2.00T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 - 0.925T + 31T^{2} \) |
| 37 | \( 1 - 7.11T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 0.434T + 43T^{2} \) |
| 47 | \( 1 + 4.07T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 9.25T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 4.78T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 4.97T + 83T^{2} \) |
| 89 | \( 1 - 1.11T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−7.71502605002561145630822762578, −6.47818472600026679868226984561, −6.05996952342658029077938402370, −5.47575735816145112051103072536, −4.65073399048056121059934527455, −4.02300091344174407442288322627, −3.10961678383349361865142929441, −2.29477381799434900303217378101, −1.21869417167546285589605540564, 0,
1.21869417167546285589605540564, 2.29477381799434900303217378101, 3.10961678383349361865142929441, 4.02300091344174407442288322627, 4.65073399048056121059934527455, 5.47575735816145112051103072536, 6.05996952342658029077938402370, 6.47818472600026679868226984561, 7.71502605002561145630822762578
