| L(s) = 1 | − 2.27·2-s − 0.713·3-s + 3.19·4-s − 3.36·5-s + 1.62·6-s − 2.72·8-s − 2.49·9-s + 7.66·10-s − 2.45·11-s − 2.27·12-s + 1.53·13-s + 2.40·15-s − 0.186·16-s − 2.89·17-s + 5.67·18-s + 1.01·19-s − 10.7·20-s + 5.59·22-s − 4.93·23-s + 1.94·24-s + 6.32·25-s − 3.48·26-s + 3.91·27-s − 8.76·29-s − 5.47·30-s − 5.91·31-s + 5.86·32-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 0.412·3-s + 1.59·4-s − 1.50·5-s + 0.663·6-s − 0.962·8-s − 0.830·9-s + 2.42·10-s − 0.740·11-s − 0.658·12-s + 0.424·13-s + 0.620·15-s − 0.0465·16-s − 0.702·17-s + 1.33·18-s + 0.233·19-s − 2.40·20-s + 1.19·22-s − 1.02·23-s + 0.396·24-s + 1.26·25-s − 0.684·26-s + 0.754·27-s − 1.62·29-s − 0.999·30-s − 1.06·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06085376931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06085376931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 + 0.713T + 3T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + 7.81T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 9.20T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 3.80T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−8.809616059454689569470783469819, −8.552731323115980619969233721317, −7.63917701267818791246470503572, −7.32250744817146815175546220861, −6.25458323334792538954967815878, −5.27592498071854092468712944240, −4.11189136792980199081260968474, −3.12028660529374622392870099643, −1.83532163463131712098403171849, −0.20361237151937738823767285487,
0.20361237151937738823767285487, 1.83532163463131712098403171849, 3.12028660529374622392870099643, 4.11189136792980199081260968474, 5.27592498071854092468712944240, 6.25458323334792538954967815878, 7.32250744817146815175546220861, 7.63917701267818791246470503572, 8.552731323115980619969233721317, 8.809616059454689569470783469819
