| L(s) = 1 | − 2-s − 0.0653·3-s + 4-s + 0.997·5-s + 0.0653·6-s + 0.699·7-s − 8-s − 2.99·9-s − 0.997·10-s + 5.54·11-s − 0.0653·12-s + 4.90·13-s − 0.699·14-s − 0.0651·15-s + 16-s + 17-s + 2.99·18-s + 1.04·19-s + 0.997·20-s − 0.0456·21-s − 5.54·22-s + 4.54·23-s + 0.0653·24-s − 4.00·25-s − 4.90·26-s + 0.391·27-s + 0.699·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0377·3-s + 0.5·4-s + 0.446·5-s + 0.0266·6-s + 0.264·7-s − 0.353·8-s − 0.998·9-s − 0.315·10-s + 1.67·11-s − 0.0188·12-s + 1.36·13-s − 0.186·14-s − 0.0168·15-s + 0.250·16-s + 0.242·17-s + 0.706·18-s + 0.240·19-s + 0.223·20-s − 0.00996·21-s − 1.18·22-s + 0.947·23-s + 0.0133·24-s − 0.800·25-s − 0.962·26-s + 0.0753·27-s + 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552867796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552867796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0653T + 3T^{2} \) |
| 5 | \( 1 - 0.997T + 5T^{2} \) |
| 7 | \( 1 - 0.699T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 4.90T + 13T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 + 0.548T + 53T^{2} \) |
| 61 | \( 1 - 8.29T + 61T^{2} \) |
| 67 | \( 1 + 3.27T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 - 6.00T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 7.47T + 97T^{2} \) |
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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
−9.099211545020290242373291514028, −8.615552757870561926046956826744, −7.72490071729010602194550465104, −6.76794743593420929704985323859, −6.05118454165613733623388882780, −5.47693508118889762697105651920, −4.03364067852954431205617843574, −3.24575866158317184736907205889, −1.92874544872067782161921763383, −0.987757239435463642280214039375,
0.987757239435463642280214039375, 1.92874544872067782161921763383, 3.24575866158317184736907205889, 4.03364067852954431205617843574, 5.47693508118889762697105651920, 6.05118454165613733623388882780, 6.76794743593420929704985323859, 7.72490071729010602194550465104, 8.615552757870561926046956826744, 9.099211545020290242373291514028
