| L(s) = 1 | − 4.05·2-s + 2.41·3-s + 8.45·4-s + 20.0·5-s − 9.80·6-s + 7·7-s − 1.83·8-s − 21.1·9-s − 81.5·10-s + 1.19·11-s + 20.4·12-s + 13.3·13-s − 28.3·14-s + 48.5·15-s − 60.1·16-s + 17·17-s + 85.8·18-s + 71.4·19-s + 169.·20-s + 16.9·21-s − 4.84·22-s + 50.4·23-s − 4.42·24-s + 278.·25-s − 54.3·26-s − 116.·27-s + 59.1·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 0.465·3-s + 1.05·4-s + 1.79·5-s − 0.667·6-s + 0.377·7-s − 0.0809·8-s − 0.783·9-s − 2.57·10-s + 0.0327·11-s + 0.491·12-s + 0.285·13-s − 0.542·14-s + 0.836·15-s − 0.940·16-s + 0.242·17-s + 1.12·18-s + 0.862·19-s + 1.89·20-s + 0.175·21-s − 0.0469·22-s + 0.457·23-s − 0.0376·24-s + 2.23·25-s − 0.409·26-s − 0.829·27-s + 0.399·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.247828743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247828743\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 4.05T + 8T^{2} \) |
| 3 | \( 1 - 2.41T + 27T^{2} \) |
| 5 | \( 1 - 20.0T + 125T^{2} \) |
| 11 | \( 1 - 1.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 71.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 231.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 11.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 362.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 517.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 395.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 479.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 717.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 286.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 498.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 719.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 734.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.37278177814917348958076315472, −11.58947862127119544862590227292, −10.46762987823706809906756466873, −9.640253901848555989278127065207, −8.937547221278431864672249803925, −7.995260818198155922894391388038, −6.54592577706183469670393730802, −5.28738202607810382957629486281, −2.63221884268312863606032404417, −1.33353272187663302917383706765,
1.33353272187663302917383706765, 2.63221884268312863606032404417, 5.28738202607810382957629486281, 6.54592577706183469670393730802, 7.995260818198155922894391388038, 8.937547221278431864672249803925, 9.640253901848555989278127065207, 10.46762987823706809906756466873, 11.58947862127119544862590227292, 13.37278177814917348958076315472
