| L(s) = 1 | − 2.23·2-s + 1.23·3-s + 3.00·4-s + 2·5-s − 2.76·6-s − 2.23·8-s − 1.47·9-s − 4.47·10-s + 3.70·12-s + 3.23·13-s + 2.47·15-s − 0.999·16-s − 3.23·17-s + 3.29·18-s + 6.47·19-s + 6.00·20-s + 2.47·23-s − 2.76·24-s − 25-s − 7.23·26-s − 5.52·27-s − 8.47·29-s − 5.52·30-s + 2.76·31-s + 6.70·32-s + 7.23·34-s − 4.41·36-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.713·3-s + 1.50·4-s + 0.894·5-s − 1.12·6-s − 0.790·8-s − 0.490·9-s − 1.41·10-s + 1.07·12-s + 0.897·13-s + 0.638·15-s − 0.249·16-s − 0.784·17-s + 0.775·18-s + 1.48·19-s + 1.34·20-s + 0.515·23-s − 0.564·24-s − 0.200·25-s − 1.41·26-s − 1.06·27-s − 1.57·29-s − 1.00·30-s + 0.496·31-s + 1.18·32-s + 1.24·34-s − 0.736·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 0.763T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.054877621056511476610374490863, −7.20856842184803188154823380870, −6.64387989946287276044841996166, −5.75726493465131887173412176812, −5.06833097811964025503399900566, −3.67097451479306978104885672307, −2.95033098057096663275561084504, −1.94683675547578347683787847024, −1.44469945725688672209077765858, 0,
1.44469945725688672209077765858, 1.94683675547578347683787847024, 2.95033098057096663275561084504, 3.67097451479306978104885672307, 5.06833097811964025503399900566, 5.75726493465131887173412176812, 6.64387989946287276044841996166, 7.20856842184803188154823380870, 8.054877621056511476610374490863
