| L(s) = 1 | − 2.23·2-s + 3.01·4-s − 0.715·5-s − 7-s − 2.26·8-s + 1.60·10-s − 3.48·11-s − 3.88·13-s + 2.23·14-s − 0.957·16-s + 2.02·17-s + 1.32·19-s − 2.15·20-s + 7.80·22-s + 4.77·23-s − 4.48·25-s + 8.68·26-s − 3.01·28-s + 8.26·29-s − 4.73·31-s + 6.66·32-s − 4.52·34-s + 0.715·35-s − 10.4·37-s − 2.96·38-s + 1.61·40-s − 0.827·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.319·5-s − 0.377·7-s − 0.799·8-s + 0.506·10-s − 1.05·11-s − 1.07·13-s + 0.598·14-s − 0.239·16-s + 0.490·17-s + 0.304·19-s − 0.481·20-s + 1.66·22-s + 0.995·23-s − 0.897·25-s + 1.70·26-s − 0.568·28-s + 1.53·29-s − 0.849·31-s + 1.17·32-s − 0.775·34-s + 0.120·35-s − 1.72·37-s − 0.481·38-s + 0.255·40-s − 0.129·41-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 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 + 0.715T + 5T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 8.26T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 0.827T + 41T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 6.64T + 59T^{2} \) |
| 61 | \( 1 - 9.79T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 8.80T + 71T^{2} \) |
| 73 | \( 1 - 6.52T + 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 - 6.83T + 97T^{2} \) |
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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
−7.50885558052859105898213437104, −7.22862176030806295847412095873, −6.45979438989052394553429298302, −5.39300269664693002879325464414, −4.86210849763590330644142844741, −3.68144652372385692456834743041, −2.75995491568647800951923980050, −2.10974784834497625438596081535, −0.897280129206787925324532153400, 0,
0.897280129206787925324532153400, 2.10974784834497625438596081535, 2.75995491568647800951923980050, 3.68144652372385692456834743041, 4.86210849763590330644142844741, 5.39300269664693002879325464414, 6.45979438989052394553429298302, 7.22862176030806295847412095873, 7.50885558052859105898213437104
