| L(s) = 1 | − 1.64·2-s + 0.702·4-s − 1.50·5-s − 7-s + 2.13·8-s + 2.46·10-s − 3.11·11-s + 0.853·13-s + 1.64·14-s − 4.91·16-s − 2.49·17-s + 7.85·19-s − 1.05·20-s + 5.11·22-s − 1.92·23-s − 2.74·25-s − 1.40·26-s − 0.702·28-s − 4.22·29-s + 10.7·31-s + 3.80·32-s + 4.10·34-s + 1.50·35-s − 9.99·37-s − 12.9·38-s − 3.20·40-s − 0.296·41-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.351·4-s − 0.671·5-s − 0.377·7-s + 0.754·8-s + 0.781·10-s − 0.938·11-s + 0.236·13-s + 0.439·14-s − 1.22·16-s − 0.605·17-s + 1.80·19-s − 0.236·20-s + 1.09·22-s − 0.401·23-s − 0.548·25-s − 0.275·26-s − 0.132·28-s − 0.785·29-s + 1.92·31-s + 0.673·32-s + 0.703·34-s + 0.253·35-s − 1.64·37-s − 2.09·38-s − 0.506·40-s − 0.0463·41-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 | \( 1 + 1.64T + 2T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 0.853T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 7.85T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 9.99T + 37T^{2} \) |
| 41 | \( 1 + 0.296T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 + 5.87T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 + 0.392T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 5.04T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.60568142667114536000156651883, −7.21039115724257145467530516879, −6.28573973488737332314190378016, −5.38559157164993712699876762278, −4.67712936607382226866974306923, −3.83385644074575684750610007443, −3.01855697012249763199243976515, −2.03280165861727823148288859277, −0.910878512888599325880069065875, 0,
0.910878512888599325880069065875, 2.03280165861727823148288859277, 3.01855697012249763199243976515, 3.83385644074575684750610007443, 4.67712936607382226866974306923, 5.38559157164993712699876762278, 6.28573973488737332314190378016, 7.21039115724257145467530516879, 7.60568142667114536000156651883
