L(s) = 1 | + (−0.577 − 0.816i)2-s + (−0.222 − 0.974i)3-s + (−0.332 + 0.942i)4-s + (−0.244 − 0.969i)5-s + (−0.667 + 0.744i)6-s + (0.548 + 0.835i)7-s + (0.962 − 0.272i)8-s + (−0.900 + 0.433i)9-s + (−0.650 + 0.759i)10-s + (−0.632 − 0.774i)11-s + (0.993 + 0.114i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.890 + 0.454i)15-s + (−0.778 − 0.627i)16-s + (−0.958 + 0.283i)17-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)2-s + (−0.222 − 0.974i)3-s + (−0.332 + 0.942i)4-s + (−0.244 − 0.969i)5-s + (−0.667 + 0.744i)6-s + (0.548 + 0.835i)7-s + (0.962 − 0.272i)8-s + (−0.900 + 0.433i)9-s + (−0.650 + 0.759i)10-s + (−0.632 − 0.774i)11-s + (0.993 + 0.114i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.890 + 0.454i)15-s + (−0.778 − 0.627i)16-s + (−0.958 + 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1423377111 + 0.05707618456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1423377111 + 0.05707618456i\) |
\(L(1)\) |
\(\approx\) |
\(0.4250172873 - 0.3097955466i\) |
\(L(1)\) |
\(\approx\) |
\(0.4250172873 - 0.3097955466i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (-0.577 - 0.816i)T \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (-0.244 - 0.969i)T \) |
| 7 | \( 1 + (0.548 + 0.835i)T \) |
| 11 | \( 1 + (-0.632 - 0.774i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (-0.958 + 0.283i)T \) |
| 19 | \( 1 + (0.973 - 0.228i)T \) |
| 23 | \( 1 + (0.968 + 0.250i)T \) |
| 29 | \( 1 + (-0.539 + 0.842i)T \) |
| 31 | \( 1 + (-0.928 - 0.370i)T \) |
| 37 | \( 1 + (-0.980 - 0.194i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.965 - 0.261i)T \) |
| 47 | \( 1 + (-0.919 + 0.391i)T \) |
| 53 | \( 1 + (0.838 + 0.544i)T \) |
| 59 | \( 1 + (0.428 + 0.903i)T \) |
| 61 | \( 1 + (0.166 + 0.986i)T \) |
| 67 | \( 1 + (-0.397 - 0.917i)T \) |
| 71 | \( 1 + (0.863 - 0.504i)T \) |
| 73 | \( 1 + (-0.890 - 0.454i)T \) |
| 79 | \( 1 + (0.256 + 0.966i)T \) |
| 83 | \( 1 + (0.948 - 0.316i)T \) |
| 89 | \( 1 + (0.770 + 0.636i)T \) |
| 97 | \( 1 + (-0.999 - 0.0115i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.102368859451023099142421936820, −22.722671237291428370338099536618, −21.86147526263377801290657861063, −20.527629772601619682421828226000, −20.03571815260053099857060968811, −18.935003592772382999355778838660, −17.827942176505273761336843092408, −17.4908738727690802082771197789, −16.5083559473360863174337152529, −15.59441562594384152767452354845, −14.942664742386494203334140464479, −14.403847904592748060041624966053, −13.39372656787074223574499096809, −11.67313545422538436446988245875, −10.86532906947928934614945871048, −10.18028278060887785291909900618, −9.57427816633686495831839868041, −8.31012355558129819287474345711, −7.30443098517295134017279933163, −6.81546074212828654823279850936, −5.301858683155993059014507218346, −4.7593658413270220325551778081, −3.58871386054228491607937651251, −2.14873111490269716010869897196, −0.10344477817512687611534148752,
1.28702737640486427131152765848, 2.174830747045116800548191775689, 3.21261778868294518119344358713, 4.869335283356635563969513471586, 5.44692636133640741094430964475, 7.0857382496165100380667132100, 7.921750217747302030534020679432, 8.70233203801317516399788567462, 9.29119380860608733614632083363, 10.83598776214844925027036016043, 11.57706356571372490531561368535, 12.1830231623350725536106217751, 13.02694051877225721921353873097, 13.590003131167322914132170825650, 15.00066591293026229184541541476, 16.27270044458366086031409599884, 16.94926704938286632437637114463, 17.869569597106995596525771914496, 18.4173180608917263387509981110, 19.36099768324008598357494027867, 19.91382955387821010854074965758, 20.83681947517054811568680906600, 21.71627971378166853035000027626, 22.42079708975193200757599749779, 23.71172355251914230302760618911