| L(s) = 1 | + 2.12·2-s + 2.50·4-s + 2.22·5-s + 7-s + 1.08·8-s + 4.73·10-s − 3.48·11-s − 5.41·13-s + 2.12·14-s − 2.72·16-s − 2.65·17-s − 5.06·19-s + 5.59·20-s − 7.39·22-s + 3.41·23-s − 0.0299·25-s − 11.4·26-s + 2.50·28-s + 0.715·29-s + 6.85·31-s − 7.94·32-s − 5.63·34-s + 2.22·35-s − 7.20·37-s − 10.7·38-s + 2.41·40-s − 3.98·41-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.25·4-s + 0.996·5-s + 0.377·7-s + 0.382·8-s + 1.49·10-s − 1.04·11-s − 1.50·13-s + 0.567·14-s − 0.680·16-s − 0.643·17-s − 1.16·19-s + 1.25·20-s − 1.57·22-s + 0.712·23-s − 0.00599·25-s − 2.25·26-s + 0.474·28-s + 0.132·29-s + 1.23·31-s − 1.40·32-s − 0.965·34-s + 0.376·35-s − 1.18·37-s − 1.74·38-s + 0.381·40-s − 0.622·41-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 | \( 1 - 2.12T + 2T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 2.65T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 - 0.715T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 + 7.20T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 4.05T + 43T^{2} \) |
| 47 | \( 1 + 3.64T + 47T^{2} \) |
| 53 | \( 1 - 0.167T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 + 7.89T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 3.57T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.04896994119862378738384498490, −6.71247581781598744384657598088, −5.84750878620965194986119300730, −5.20450800126890609870078903078, −4.84263270787284256231040403177, −4.16191669214642448589010701747, −3.00394985237531227056206959751, −2.44064967593495814589242224570, −1.85986417963948528184789806517, 0,
1.85986417963948528184789806517, 2.44064967593495814589242224570, 3.00394985237531227056206959751, 4.16191669214642448589010701747, 4.84263270787284256231040403177, 5.20450800126890609870078903078, 5.84750878620965194986119300730, 6.71247581781598744384657598088, 7.04896994119862378738384498490
