L(s) = 1 | + (0.438 + 0.898i)2-s + (0.961 − 0.275i)3-s + (−0.615 + 0.788i)4-s + (−0.104 − 0.994i)5-s + (0.669 + 0.743i)6-s + (−0.5 − 0.866i)7-s + (−0.978 − 0.207i)8-s + (0.848 − 0.529i)9-s + (0.848 − 0.529i)10-s + (0.990 + 0.139i)11-s + (−0.374 + 0.927i)12-s + (0.559 + 0.829i)13-s + (0.559 − 0.829i)14-s + (−0.374 − 0.927i)15-s + (−0.241 − 0.970i)16-s + (0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (0.438 + 0.898i)2-s + (0.961 − 0.275i)3-s + (−0.615 + 0.788i)4-s + (−0.104 − 0.994i)5-s + (0.669 + 0.743i)6-s + (−0.5 − 0.866i)7-s + (−0.978 − 0.207i)8-s + (0.848 − 0.529i)9-s + (0.848 − 0.529i)10-s + (0.990 + 0.139i)11-s + (−0.374 + 0.927i)12-s + (0.559 + 0.829i)13-s + (0.559 − 0.829i)14-s + (−0.374 − 0.927i)15-s + (−0.241 − 0.970i)16-s + (0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.743468889 + 0.3293730306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.743468889 + 0.3293730306i\) |
\(L(1)\) |
\(\approx\) |
\(1.544392094 + 0.3165229763i\) |
\(L(1)\) |
\(\approx\) |
\(1.544392094 + 0.3165229763i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 181 | \( 1 \) |
good | 2 | \( 1 + (0.438 + 0.898i)T \) |
| 3 | \( 1 + (0.961 - 0.275i)T \) |
| 5 | \( 1 + (-0.104 - 0.994i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.990 + 0.139i)T \) |
| 13 | \( 1 + (0.559 + 0.829i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.882 + 0.469i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.719 - 0.694i)T \) |
| 41 | \( 1 + (0.0348 - 0.999i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.374 + 0.927i)T \) |
| 53 | \( 1 + (0.0348 + 0.999i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.669 + 0.743i)T \) |
| 71 | \( 1 + (0.913 - 0.406i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.0348 + 0.999i)T \) |
| 83 | \( 1 + (-0.997 - 0.0697i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.990 - 0.139i)T \) |
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
−27.40269697736852455108037357977, −26.38660892013969219351063988040, −25.36669013440983802469924706362, −24.52160995921544706962902964854, −22.95558173029306761998480818610, −22.230956121956090825703655171732, −21.6199865395517180113079354715, −20.36496430111582053162259890491, −19.71049339533979854743294793805, −18.6133699808624040015782570415, −18.26059653522022803362928372089, −16.09493219915871211774503786452, −15.07180548644739707310080821705, −14.38385047276391200616113392228, −13.5102029091016345733407795668, −12.29992754567583651522713152844, −11.29081889848376885969631120354, −10.0460773316567075482911208297, −9.39882596236734705924565296173, −8.16734633602562419527200244271, −6.57660620630165009493136855602, −5.28530666681383924856771721399, −3.47163109415724122641793593553, −3.229853444378677997964440501037, −1.83711397805137401672094848158,
1.43325519121453594065830172118, 3.71752953983274921647553135820, 4.04231809554654458269450778447, 5.77149988910322619618251413311, 7.03812918549436792378758486846, 7.86334355598069031260579329314, 9.01227299357757244003405512556, 9.65228870621324422957112647375, 11.889074669085281701399326349851, 12.82647977604676974851416177895, 13.73568668318512110427871306035, 14.32437053904846742240004043832, 15.645633419816198370898748144625, 16.48891414544482849536402232297, 17.26752154216218474370805505591, 18.666744423962204177737352505988, 19.78733548088022472871691676215, 20.58322576245134860698901962095, 21.569854196648779903044250522066, 22.83851667178742910160866341811, 23.95496261664011068151633097559, 24.32274179923427654567655145938, 25.51944636551001462631968314812, 26.03203114100583072948530673984, 27.053583699278151805690013597598