| L(s) = 1 | − 2.76·2-s + 5.65·4-s + 3.65·5-s − 7-s − 10.1·8-s − 10.1·10-s − 0.555·11-s + 1.94·13-s + 2.76·14-s + 16.6·16-s − 6.99·17-s − 2.93·19-s + 20.6·20-s + 1.53·22-s + 4.18·23-s + 8.35·25-s − 5.39·26-s − 5.65·28-s − 8.74·29-s + 0.936·31-s − 25.9·32-s + 19.3·34-s − 3.65·35-s + 2.96·37-s + 8.12·38-s − 36.9·40-s + 0.248·41-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 2.82·4-s + 1.63·5-s − 0.377·7-s − 3.57·8-s − 3.19·10-s − 0.167·11-s + 0.540·13-s + 0.739·14-s + 4.17·16-s − 1.69·17-s − 0.673·19-s + 4.62·20-s + 0.327·22-s + 0.872·23-s + 1.67·25-s − 1.05·26-s − 1.06·28-s − 1.62·29-s + 0.168·31-s − 4.58·32-s + 3.31·34-s − 0.617·35-s + 0.486·37-s + 1.31·38-s − 5.84·40-s + 0.0387·41-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 | \( 1 + 2.76T + 2T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 11 | \( 1 + 0.555T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 0.936T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 - 0.248T + 41T^{2} \) |
| 43 | \( 1 - 0.630T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + 3.50T + 59T^{2} \) |
| 61 | \( 1 - 0.0482T + 61T^{2} \) |
| 67 | \( 1 - 1.92T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 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
−7.56554619649722185401269271626, −6.83708145783465885178696709136, −6.30167071455970294425502107785, −5.95327312523420868141245620084, −4.93327501365447299577717791516, −3.47804729304043837410304807321, −2.45614259047245766862569958900, −2.06720410513017576848885088720, −1.22872808577451624965254824632, 0,
1.22872808577451624965254824632, 2.06720410513017576848885088720, 2.45614259047245766862569958900, 3.47804729304043837410304807321, 4.93327501365447299577717791516, 5.95327312523420868141245620084, 6.30167071455970294425502107785, 6.83708145783465885178696709136, 7.56554619649722185401269271626
