| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 4i·7-s − i·8-s − 9-s + 4·11-s + i·12-s − i·13-s + 4·14-s + 16-s − 4i·17-s − i·18-s − 7·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.277i·13-s + 1.06·14-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s − 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082723953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082723953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
| 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
−8.670231061257909564688716669936, −8.070356942631030067294721460575, −7.17679489994874949518618585846, −6.68822946218368444751011820651, −6.11767895961085587848521711702, −4.73570751545750565196748619118, −4.20544418149630622183628517188, −3.12536449962335521660315023991, −1.53644023203170128004644993958, −0.39435880015114462270059385706,
1.70879922553749039211126270533, 2.48870016853718065610658044961, 3.75309334582671404560127280816, 4.22718074088166415958433641186, 5.49039042451298610184716050483, 5.95894365267097533067131892314, 6.99470142407571856554613755995, 8.459272151221624133241446892714, 8.739592408247083859082926066477, 9.384106465165423691845979452372
