| L(s) = 1 | + 1.86·2-s + 1.47·4-s + 1.88·5-s + 7-s − 0.976·8-s + 3.50·10-s − 3.49·11-s + 1.17·13-s + 1.86·14-s − 4.77·16-s + 3.53·17-s − 4.46·19-s + 2.77·20-s − 6.52·22-s + 0.125·23-s − 1.45·25-s + 2.19·26-s + 1.47·28-s − 9.36·29-s − 10.0·31-s − 6.94·32-s + 6.59·34-s + 1.88·35-s + 2.41·37-s − 8.32·38-s − 1.83·40-s − 5.72·41-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 0.737·4-s + 0.841·5-s + 0.377·7-s − 0.345·8-s + 1.10·10-s − 1.05·11-s + 0.325·13-s + 0.498·14-s − 1.19·16-s + 0.857·17-s − 1.02·19-s + 0.621·20-s − 1.39·22-s + 0.0262·23-s − 0.291·25-s + 0.429·26-s + 0.278·28-s − 1.73·29-s − 1.79·31-s − 1.22·32-s + 1.13·34-s + 0.318·35-s + 0.397·37-s − 1.35·38-s − 0.290·40-s − 0.894·41-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 | \( 1 - 1.86T + 2T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 0.125T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 + 5.72T + 41T^{2} \) |
| 43 | \( 1 + 0.132T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 - 7.57T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 + 7.88T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.41435975598745725654100100601, −6.45318998165537253536643523093, −5.84142060071552074025799315067, −5.37348541864064697522164258869, −4.85335469148705033662537396986, −3.87539485718714853579508056653, −3.33100284078435764714538247055, −2.30282455283813094785613613769, −1.75165894168644963281739209349, 0,
1.75165894168644963281739209349, 2.30282455283813094785613613769, 3.33100284078435764714538247055, 3.87539485718714853579508056653, 4.85335469148705033662537396986, 5.37348541864064697522164258869, 5.84142060071552074025799315067, 6.45318998165537253536643523093, 7.41435975598745725654100100601
