L(s) = 1 | + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (0.939 − 0.342i)5-s + (−0.719 − 0.694i)7-s + (0.104 − 0.994i)8-s + (0.669 − 0.743i)10-s + (0.719 + 0.694i)11-s + (0.848 − 0.529i)13-s + (−0.961 − 0.275i)14-s + (−0.374 − 0.927i)16-s + (0.809 + 0.587i)17-s + (−0.978 − 0.207i)19-s + (0.241 − 0.970i)20-s + (0.961 + 0.275i)22-s + (−0.961 − 0.275i)23-s + ⋯ |
L(s) = 1 | + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (0.939 − 0.342i)5-s + (−0.719 − 0.694i)7-s + (0.104 − 0.994i)8-s + (0.669 − 0.743i)10-s + (0.719 + 0.694i)11-s + (0.848 − 0.529i)13-s + (−0.961 − 0.275i)14-s + (−0.374 − 0.927i)16-s + (0.809 + 0.587i)17-s + (−0.978 − 0.207i)19-s + (0.241 − 0.970i)20-s + (0.961 + 0.275i)22-s + (−0.961 − 0.275i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.914828515 - 4.048495524i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.914828515 - 4.048495524i\) |
\(L(1)\) |
\(\approx\) |
\(1.764813758 - 1.161441749i\) |
\(L(1)\) |
\(\approx\) |
\(1.764813758 - 1.161441749i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.882 - 0.469i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.719 - 0.694i)T \) |
| 11 | \( 1 + (0.719 + 0.694i)T \) |
| 13 | \( 1 + (0.848 - 0.529i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.961 - 0.275i)T \) |
| 29 | \( 1 + (0.882 - 0.469i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.374 - 0.927i)T \) |
| 43 | \( 1 + (0.0348 - 0.999i)T \) |
| 47 | \( 1 + (0.615 - 0.788i)T \) |
| 53 | \( 1 + (-0.913 + 0.406i)T \) |
| 59 | \( 1 + (0.882 + 0.469i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.104 - 0.994i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.997 + 0.0697i)T \) |
| 83 | \( 1 + (-0.0348 + 0.999i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.241 + 0.970i)T \) |
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
−22.15922107650051974101593882799, −21.537430769710228894781755191495, −21.12857355530811935122636369005, −19.95730858181889247330637392258, −18.96020369677700239977448642636, −18.20493636281423424467355694479, −17.238126147539077875348926250429, −16.33196396491029580275977247432, −15.96221058217821191000343541940, −14.66954386171588140715999896308, −14.23071483307708987550410293145, −13.41380813123315844533399785001, −12.67932723762589361492195869274, −11.77434140838505744511049526058, −10.97703575841310079844907150984, −9.78866450533702765410247566349, −8.95942894715983687556413529966, −8.05238936356017720447064302537, −6.686947171345997030817966308224, −6.18842796990639595062825216908, −5.67305486701163628539341523014, −4.39408553982655369734253149610, −3.33854802411379995665100955621, −2.61834035803842044358600798553, −1.44006491952067675768355336651,
0.68919396817724530034555963065, 1.61903767282540059798967134936, 2.63215733592572911589816516599, 3.823699251711898497801424407553, 4.39981799468411557363371575528, 5.73057509560601108090921570322, 6.224437775909780654695173521179, 7.080299748437882898940024821537, 8.48873321818854214498537732833, 9.65837017225965378771240013041, 10.19446225585606239004697403627, 10.88609021543401746146738696874, 12.254635292629042372320439007672, 12.67313450495822890027147504956, 13.55781958927925487622087012092, 14.096059768250097482890788903151, 15.01211103343669887323955639460, 15.95592304062737144385626804380, 16.82044919785775207898259109657, 17.54435142856806256692729244211, 18.64737004096776916560125579428, 19.55705490512856458220559428507, 20.25560512196894993974101405606, 20.84234395331597136546451170728, 21.72015112640481082137183687018