| L(s) = 1 | + 0.384·2-s − 1.85·4-s − 3.33·5-s + 7-s − 1.48·8-s − 1.28·10-s − 0.515·11-s + 0.383·13-s + 0.384·14-s + 3.13·16-s − 5.06·17-s − 3.66·19-s + 6.17·20-s − 0.198·22-s + 5.75·23-s + 6.12·25-s + 0.147·26-s − 1.85·28-s + 5.53·29-s − 1.26·31-s + 4.16·32-s − 1.94·34-s − 3.33·35-s − 1.25·37-s − 1.40·38-s + 4.94·40-s + 3.87·41-s + ⋯ |
| L(s) = 1 | + 0.271·2-s − 0.926·4-s − 1.49·5-s + 0.377·7-s − 0.523·8-s − 0.405·10-s − 0.155·11-s + 0.106·13-s + 0.102·14-s + 0.783·16-s − 1.22·17-s − 0.840·19-s + 1.38·20-s − 0.0422·22-s + 1.20·23-s + 1.22·25-s + 0.0289·26-s − 0.350·28-s + 1.02·29-s − 0.226·31-s + 0.736·32-s − 0.334·34-s − 0.563·35-s − 0.206·37-s − 0.228·38-s + 0.781·40-s + 0.604·41-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 | \( 1 - 0.384T + 2T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 11 | \( 1 + 0.515T + 11T^{2} \) |
| 13 | \( 1 - 0.383T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 - 5.75T + 23T^{2} \) |
| 29 | \( 1 - 5.53T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 - 3.87T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.05T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 - 8.42T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.48279839561981237300453560464, −6.95424763577452655882127649063, −6.06277620047204426953445886997, −5.09318476742149573784318798431, −4.54732534462271393861898267534, −4.07287024810035394625973823723, −3.35348004320702806606024943521, −2.42151694585635320629548487798, −0.943510042622814131818536097598, 0,
0.943510042622814131818536097598, 2.42151694585635320629548487798, 3.35348004320702806606024943521, 4.07287024810035394625973823723, 4.54732534462271393861898267534, 5.09318476742149573784318798431, 6.06277620047204426953445886997, 6.95424763577452655882127649063, 7.48279839561981237300453560464
