| L(s) = 1 | − 0.481·2-s + 0.287·3-s − 1.76·4-s − 5-s − 0.138·6-s − 4.64·7-s + 1.81·8-s − 2.91·9-s + 0.481·10-s + 4.58·11-s − 0.508·12-s − 6.19·13-s + 2.23·14-s − 0.287·15-s + 2.66·16-s − 6.60·17-s + 1.40·18-s − 2.42·19-s + 1.76·20-s − 1.33·21-s − 2.20·22-s − 5.56·23-s + 0.521·24-s + 25-s + 2.97·26-s − 1.70·27-s + 8.22·28-s + ⋯ |
| L(s) = 1 | − 0.340·2-s + 0.165·3-s − 0.884·4-s − 0.447·5-s − 0.0564·6-s − 1.75·7-s + 0.641·8-s − 0.972·9-s + 0.152·10-s + 1.38·11-s − 0.146·12-s − 1.71·13-s + 0.597·14-s − 0.0741·15-s + 0.666·16-s − 1.60·17-s + 0.330·18-s − 0.556·19-s + 0.395·20-s − 0.291·21-s − 0.470·22-s − 1.16·23-s + 0.106·24-s + 0.200·25-s + 0.584·26-s − 0.327·27-s + 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2843866352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2843866352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.481T + 2T^{2} \) |
| 3 | \( 1 - 0.287T + 3T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 0.894T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 + 7.13T + 41T^{2} \) |
| 43 | \( 1 - 6.36T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 0.0507T + 53T^{2} \) |
| 59 | \( 1 - 0.810T + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 + 6.05T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 0.720T + 83T^{2} \) |
| 89 | \( 1 + 0.586T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.967087047244352203666361795416, −8.754183942323162181776640038785, −7.71752173122299580577885779114, −6.61830545046183032124936964556, −6.33982896907113855767488733351, −4.94064664023000118141105583178, −4.18190961595441630083163883338, −3.35571539340459388645929544535, −2.34671594100718775514099015001, −0.34834019735302982163469150687,
0.34834019735302982163469150687, 2.34671594100718775514099015001, 3.35571539340459388645929544535, 4.18190961595441630083163883338, 4.94064664023000118141105583178, 6.33982896907113855767488733351, 6.61830545046183032124936964556, 7.71752173122299580577885779114, 8.754183942323162181776640038785, 8.967087047244352203666361795416
