| L(s) = 1 | + 2.08·2-s + 2.34·4-s + 0.447·5-s − 1.20·7-s + 0.729·8-s + 0.932·10-s − 11-s − 1.30·13-s − 2.50·14-s − 3.17·16-s − 0.935·17-s + 8.50·19-s + 1.05·20-s − 2.08·22-s − 5.29·23-s − 4.80·25-s − 2.72·26-s − 2.82·28-s − 0.0184·29-s + 0.795·31-s − 8.08·32-s − 1.95·34-s − 0.537·35-s − 2.49·37-s + 17.7·38-s + 0.326·40-s + 5.14·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s + 0.199·5-s − 0.454·7-s + 0.257·8-s + 0.294·10-s − 0.301·11-s − 0.361·13-s − 0.669·14-s − 0.794·16-s − 0.226·17-s + 1.95·19-s + 0.234·20-s − 0.444·22-s − 1.10·23-s − 0.960·25-s − 0.533·26-s − 0.533·28-s − 0.00342·29-s + 0.142·31-s − 1.42·32-s − 0.334·34-s − 0.0907·35-s − 0.409·37-s + 2.87·38-s + 0.0515·40-s + 0.804·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 0.447T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 0.935T + 17T^{2} \) |
| 19 | \( 1 - 8.50T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 0.0184T + 29T^{2} \) |
| 31 | \( 1 - 0.795T + 31T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 + 7.10T + 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 + 8.14T + 71T^{2} \) |
| 73 | \( 1 - 7.50T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 + 7.73T + 97T^{2} \) |
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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
−7.47983112056001255210704876973, −6.42636322053957592292848658543, −5.98502193694815890125411324165, −5.35866596041398224514342294664, −4.68204679474172937597788446829, −3.95960119614092870239434059574, −3.18790907186551404992180246810, −2.63938876691893935627008839057, −1.59871821641347488823884982696, 0,
1.59871821641347488823884982696, 2.63938876691893935627008839057, 3.18790907186551404992180246810, 3.95960119614092870239434059574, 4.68204679474172937597788446829, 5.35866596041398224514342294664, 5.98502193694815890125411324165, 6.42636322053957592292848658543, 7.47983112056001255210704876973
