L(s) = 1 | + (0.637 − 0.770i)2-s + (−0.876 + 0.481i)3-s + (−0.187 − 0.982i)4-s + (−0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (−0.876 − 0.481i)8-s + (0.535 − 0.844i)9-s + (0.637 + 0.770i)12-s + (−0.535 + 0.844i)13-s + (0.0627 − 0.998i)14-s + (−0.929 + 0.368i)16-s + (0.425 + 0.904i)17-s + (−0.309 − 0.951i)18-s + (−0.992 + 0.125i)19-s + (−0.425 + 0.904i)21-s + ⋯ |
L(s) = 1 | + (0.637 − 0.770i)2-s + (−0.876 + 0.481i)3-s + (−0.187 − 0.982i)4-s + (−0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (−0.876 − 0.481i)8-s + (0.535 − 0.844i)9-s + (0.637 + 0.770i)12-s + (−0.535 + 0.844i)13-s + (0.0627 − 0.998i)14-s + (−0.929 + 0.368i)16-s + (0.425 + 0.904i)17-s + (−0.309 − 0.951i)18-s + (−0.992 + 0.125i)19-s + (−0.425 + 0.904i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8111995283 + 0.4028432496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8111995283 + 0.4028432496i\) |
\(L(1)\) |
\(\approx\) |
\(0.9455981854 - 0.2164229090i\) |
\(L(1)\) |
\(\approx\) |
\(0.9455981854 - 0.2164229090i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.637 - 0.770i)T \) |
| 3 | \( 1 + (-0.876 + 0.481i)T \) |
| 7 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.535 + 0.844i)T \) |
| 17 | \( 1 + (0.425 + 0.904i)T \) |
| 19 | \( 1 + (-0.992 + 0.125i)T \) |
| 23 | \( 1 + (-0.0627 + 0.998i)T \) |
| 29 | \( 1 + (-0.425 + 0.904i)T \) |
| 31 | \( 1 + (-0.992 + 0.125i)T \) |
| 37 | \( 1 + (0.637 + 0.770i)T \) |
| 41 | \( 1 + (-0.929 + 0.368i)T \) |
| 43 | \( 1 + (-0.309 - 0.951i)T \) |
| 47 | \( 1 + (0.425 - 0.904i)T \) |
| 53 | \( 1 + (0.187 + 0.982i)T \) |
| 59 | \( 1 + (-0.929 + 0.368i)T \) |
| 61 | \( 1 + (-0.637 + 0.770i)T \) |
| 67 | \( 1 + (0.992 - 0.125i)T \) |
| 71 | \( 1 + (-0.187 - 0.982i)T \) |
| 73 | \( 1 + (-0.535 - 0.844i)T \) |
| 79 | \( 1 + (0.728 + 0.684i)T \) |
| 83 | \( 1 + (-0.728 + 0.684i)T \) |
| 89 | \( 1 + (-0.637 + 0.770i)T \) |
| 97 | \( 1 + (-0.876 + 0.481i)T \) |
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
−21.00548915037990427686936874806, −20.15695506168454215740015608394, −18.82165640996065793781115494178, −18.28963221520911692117622683490, −17.53685128175173946348816129871, −16.98643692337412689997180532787, −16.21110694663182305582556751322, −15.37309792701810058165225657548, −14.67276124544002700086823145237, −13.9496790121500309815000755740, −12.771053474299890269609741356128, −12.60837098286275970204584448623, −11.583731805235010672143308717266, −11.05972935854130386132040080838, −9.870692121709970627213922709903, −8.69183435715613611850079026879, −7.876631991463802493318410013547, −7.31788660012158847658735371037, −6.315473941583318248859058946223, −5.624351214210664029832375007172, −4.959760036211034417792837035749, −4.26805756865126372789613872686, −2.81979574944707633406521778100, −1.978246317443465790737707124319, −0.315385742793528335000834218962,
1.28225390154641981175232363466, 1.938294071137015964862858728, 3.50186140684430836058986308555, 4.11558666886774814043493476679, 4.875584698241917802655498274729, 5.57152567386437112977396442326, 6.49749458947638233163782575078, 7.363143343869862202144514437947, 8.71863809427983987485230055265, 9.60317255798349966030256255583, 10.4786774703308718670196796201, 10.8743925870384077408235225573, 11.73473273474021245412408078422, 12.269801977589025891694225489955, 13.18147985529413140872840492658, 14.05412047301743100065198082311, 14.92269397696073472597019721645, 15.27188179048928127185652968276, 16.7549483272499342655411142807, 16.91762811550827028515366211864, 18.05268198384596234318419592927, 18.664345991371780390543199038579, 19.6603658332537236128856585383, 20.339565414725885123088260881261, 21.24453634317909111328860302940