| L(s) = 1 | + 0.910·2-s − 1.17·4-s + 2.06·5-s + 7-s − 2.88·8-s + 1.88·10-s − 4.76·11-s + 2.03·13-s + 0.910·14-s − 0.290·16-s − 0.854·17-s + 2.23·19-s − 2.41·20-s − 4.33·22-s − 5.99·23-s − 0.725·25-s + 1.85·26-s − 1.17·28-s + 5.46·29-s + 3.43·31-s + 5.51·32-s − 0.778·34-s + 2.06·35-s + 1.22·37-s + 2.03·38-s − 5.97·40-s + 7.05·41-s + ⋯ |
| L(s) = 1 | + 0.644·2-s − 0.585·4-s + 0.924·5-s + 0.377·7-s − 1.02·8-s + 0.595·10-s − 1.43·11-s + 0.565·13-s + 0.243·14-s − 0.0725·16-s − 0.207·17-s + 0.513·19-s − 0.540·20-s − 0.924·22-s − 1.25·23-s − 0.145·25-s + 0.364·26-s − 0.221·28-s + 1.01·29-s + 0.616·31-s + 0.974·32-s − 0.133·34-s + 0.349·35-s + 0.201·37-s + 0.330·38-s − 0.944·40-s + 1.10·41-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 | \( 1 - 0.910T + 2T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + 0.854T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.99T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 + 5.15T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 - 1.39T + 53T^{2} \) |
| 59 | \( 1 + 8.77T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 + 5.42T + 67T^{2} \) |
| 71 | \( 1 - 1.57T + 71T^{2} \) |
| 73 | \( 1 + 9.67T + 73T^{2} \) |
| 79 | \( 1 + 4.49T + 79T^{2} \) |
| 83 | \( 1 + 8.88T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−7.61041996852531037361637846242, −6.48013172235609787259214837924, −5.90282278024165624233780655813, −5.40381317182813983453378279994, −4.73442974749308039014387478061, −4.08267028317947110412705812299, −3.05247554831402603684436475666, −2.44600041865091069018085035753, −1.36184883427356093035698038169, 0,
1.36184883427356093035698038169, 2.44600041865091069018085035753, 3.05247554831402603684436475666, 4.08267028317947110412705812299, 4.73442974749308039014387478061, 5.40381317182813983453378279994, 5.90282278024165624233780655813, 6.48013172235609787259214837924, 7.61041996852531037361637846242
