| L(s) = 1 | + 1.26·2-s + 1.16·3-s − 0.401·4-s − 0.0160·5-s + 1.47·6-s − 4.37·7-s − 3.03·8-s − 1.64·9-s − 0.0202·10-s + 0.663·11-s − 0.468·12-s + 1.57·13-s − 5.53·14-s − 0.0187·15-s − 3.03·16-s − 17-s − 2.07·18-s + 0.0656·19-s + 0.00644·20-s − 5.10·21-s + 0.839·22-s − 2.93·23-s − 3.53·24-s − 4.99·25-s + 1.99·26-s − 5.41·27-s + 1.75·28-s + ⋯ |
| L(s) = 1 | + 0.893·2-s + 0.673·3-s − 0.200·4-s − 0.00717·5-s + 0.601·6-s − 1.65·7-s − 1.07·8-s − 0.546·9-s − 0.00641·10-s + 0.200·11-s − 0.135·12-s + 0.436·13-s − 1.47·14-s − 0.00482·15-s − 0.758·16-s − 0.242·17-s − 0.488·18-s + 0.0150·19-s + 0.00144·20-s − 1.11·21-s + 0.178·22-s − 0.611·23-s − 0.722·24-s − 0.999·25-s + 0.390·26-s − 1.04·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 + 0.0160T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 - 0.663T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 19 | \( 1 - 0.0656T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + 7.47T + 53T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 7.48T + 73T^{2} \) |
| 79 | \( 1 + 0.452T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 - 9.33T + 97T^{2} \) |
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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
−9.494160293908475789460430540658, −8.830630883220489837809157078010, −8.030744763932184876275169903511, −6.65897662796367862607461422026, −6.12143017187613914378828911771, −5.20056616589185387324347662104, −3.79130489031711116920596760322, −3.47602474018700952512756425287, −2.42445625729189339060801408627, 0,
2.42445625729189339060801408627, 3.47602474018700952512756425287, 3.79130489031711116920596760322, 5.20056616589185387324347662104, 6.12143017187613914378828911771, 6.65897662796367862607461422026, 8.030744763932184876275169903511, 8.830630883220489837809157078010, 9.494160293908475789460430540658
