L(s) = 1 | + (−0.917 + 0.398i)2-s + (0.460 + 0.887i)3-s + (0.682 − 0.730i)4-s + (−0.0682 − 0.997i)5-s + (−0.775 − 0.631i)6-s + (−0.990 − 0.136i)7-s + (−0.334 + 0.942i)8-s + (−0.576 + 0.816i)9-s + (0.460 + 0.887i)10-s + (−0.576 + 0.816i)11-s + (0.962 + 0.269i)12-s + (−0.576 − 0.816i)13-s + (0.962 − 0.269i)14-s + (0.854 − 0.519i)15-s + (−0.0682 − 0.997i)16-s + (0.854 + 0.519i)17-s + ⋯ |
L(s) = 1 | + (−0.917 + 0.398i)2-s + (0.460 + 0.887i)3-s + (0.682 − 0.730i)4-s + (−0.0682 − 0.997i)5-s + (−0.775 − 0.631i)6-s + (−0.990 − 0.136i)7-s + (−0.334 + 0.942i)8-s + (−0.576 + 0.816i)9-s + (0.460 + 0.887i)10-s + (−0.576 + 0.816i)11-s + (0.962 + 0.269i)12-s + (−0.576 − 0.816i)13-s + (0.962 − 0.269i)14-s + (0.854 − 0.519i)15-s + (−0.0682 − 0.997i)16-s + (0.854 + 0.519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4816767912 - 0.2388906979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4816767912 - 0.2388906979i\) |
\(L(1)\) |
\(\approx\) |
\(0.6151125129 + 0.1108819305i\) |
\(L(1)\) |
\(\approx\) |
\(0.6151125129 + 0.1108819305i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.917 + 0.398i)T \) |
| 3 | \( 1 + (0.460 + 0.887i)T \) |
| 5 | \( 1 + (-0.0682 - 0.997i)T \) |
| 7 | \( 1 + (-0.990 - 0.136i)T \) |
| 11 | \( 1 + (-0.576 + 0.816i)T \) |
| 13 | \( 1 + (-0.576 - 0.816i)T \) |
| 17 | \( 1 + (0.854 + 0.519i)T \) |
| 19 | \( 1 + (0.460 + 0.887i)T \) |
| 23 | \( 1 + (0.682 - 0.730i)T \) |
| 29 | \( 1 + (-0.576 - 0.816i)T \) |
| 31 | \( 1 + (-0.334 + 0.942i)T \) |
| 37 | \( 1 + (-0.576 - 0.816i)T \) |
| 41 | \( 1 + (-0.775 - 0.631i)T \) |
| 43 | \( 1 + (0.854 - 0.519i)T \) |
| 47 | \( 1 + (0.854 - 0.519i)T \) |
| 53 | \( 1 + (0.682 - 0.730i)T \) |
| 59 | \( 1 + (0.682 + 0.730i)T \) |
| 61 | \( 1 + (-0.775 - 0.631i)T \) |
| 67 | \( 1 + (-0.334 - 0.942i)T \) |
| 71 | \( 1 + (-0.917 - 0.398i)T \) |
| 73 | \( 1 + (0.682 + 0.730i)T \) |
| 79 | \( 1 + (0.682 + 0.730i)T \) |
| 83 | \( 1 + (-0.775 - 0.631i)T \) |
| 89 | \( 1 + (-0.334 - 0.942i)T \) |
| 97 | \( 1 + T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.8115092989009455496283208898, −20.90906056198237184459679381633, −19.934162832644524623128120955028, −19.22973764434052937532767203466, −18.79357114048952895839193823957, −18.359273809779638712314268715952, −17.34200360885596390776552556280, −16.50548258287853733198230560570, −15.62545762013288817107087977385, −14.76219373491017793481813742720, −13.675570778856164687850538060115, −13.121769568532820660107365099905, −12.01819005126477402625295092587, −11.48345818427260900026298130481, −10.5422661564452830805681292829, −9.49275387482447146896735422441, −9.07932948839115227494542393831, −7.81728473008699326047406268114, −7.23202608064570161951026767898, −6.64326580922978183522961549855, −5.66246288328795570704700854905, −3.6136027176986780010446038449, −2.94710158459719180346787451622, −2.43873466848127929395741434377, −1.067250095622625244915820750578,
0.32636730967715044485398536947, 1.86023586262171625647741472552, 2.952064586151382672277533865317, 4.01882638177713281186515329504, 5.29304858327433663771077265434, 5.64201453743892340816060563170, 7.18625412094311068622973336579, 7.883596915315664795599706758, 8.71888665771568329320610076917, 9.4681464860880179999279083292, 10.16723378847102846123320579061, 10.54946997889676792253538040403, 12.105678875910648054158613742202, 12.70262398851908781183002286533, 13.85772998595480331942047475245, 14.885678397172339940812971791474, 15.49480486774832801425166526732, 16.17618722613214172123400458185, 16.82329715487970267751172463792, 17.365017082832788678779432509168, 18.59736800001524816819175091473, 19.42682597299246262579228367325, 20.03985266699321064313199969685, 20.656291782336037436422051505, 21.22294417399012756620179971700