L(s) = 1 | + (−0.627 + 0.778i)2-s + (0.533 + 0.845i)3-s + (−0.211 − 0.977i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (0.657 + 0.753i)7-s + (0.893 + 0.448i)8-s + (−0.431 + 0.902i)9-s + (−0.135 − 0.990i)10-s + (0.466 − 0.884i)11-s + (0.713 − 0.700i)12-s + (−0.0581 + 0.998i)13-s + (−0.999 + 0.0387i)14-s + (−0.981 − 0.192i)15-s + (−0.910 + 0.413i)16-s + (−0.835 + 0.549i)17-s + ⋯ |
L(s) = 1 | + (−0.627 + 0.778i)2-s + (0.533 + 0.845i)3-s + (−0.211 − 0.977i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (0.657 + 0.753i)7-s + (0.893 + 0.448i)8-s + (−0.431 + 0.902i)9-s + (−0.135 − 0.990i)10-s + (0.466 − 0.884i)11-s + (0.713 − 0.700i)12-s + (−0.0581 + 0.998i)13-s + (−0.999 + 0.0387i)14-s + (−0.981 − 0.192i)15-s + (−0.910 + 0.413i)16-s + (−0.835 + 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1198078430 + 0.8093565581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1198078430 + 0.8093565581i\) |
\(L(1)\) |
\(\approx\) |
\(0.5424082034 + 0.6245679630i\) |
\(L(1)\) |
\(\approx\) |
\(0.5424082034 + 0.6245679630i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 163 | \( 1 \) |
good | 2 | \( 1 + (-0.627 + 0.778i)T \) |
| 3 | \( 1 + (0.533 + 0.845i)T \) |
| 5 | \( 1 + (-0.686 + 0.727i)T \) |
| 7 | \( 1 + (0.657 + 0.753i)T \) |
| 11 | \( 1 + (0.466 - 0.884i)T \) |
| 13 | \( 1 + (-0.0581 + 0.998i)T \) |
| 17 | \( 1 + (-0.835 + 0.549i)T \) |
| 19 | \( 1 + (0.0193 - 0.999i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.360 + 0.932i)T \) |
| 31 | \( 1 + (-0.286 - 0.957i)T \) |
| 37 | \( 1 + (0.396 - 0.918i)T \) |
| 41 | \( 1 + (-0.211 + 0.977i)T \) |
| 43 | \( 1 + (-0.790 - 0.612i)T \) |
| 47 | \( 1 + (0.987 + 0.154i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.893 - 0.448i)T \) |
| 67 | \( 1 + (0.952 - 0.305i)T \) |
| 71 | \( 1 + (0.925 - 0.378i)T \) |
| 73 | \( 1 + (0.952 + 0.305i)T \) |
| 79 | \( 1 + (0.996 + 0.0774i)T \) |
| 83 | \( 1 + (0.323 - 0.946i)T \) |
| 89 | \( 1 + (0.466 + 0.884i)T \) |
| 97 | \( 1 + (-0.565 + 0.824i)T \) |
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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
−27.243142777225629620227483874678, −26.67750127903086400026846774633, −25.24238264402149358854090842080, −24.67708658250752765321367866189, −23.39265877741170870471070155122, −22.55271448108702335359493037500, −20.60774083525302090345426957820, −20.44467809831320239886211702309, −19.665300086494302419554980971211, −18.526433704209830311561778904335, −17.58954186964722693682406592984, −16.83238165576014228081272705325, −15.30598702707792070733709716059, −13.98874134585420493990537336927, −12.8549647864296614218931767296, −12.204102280061897022541115489421, −11.16498701920581421827147033101, −9.81040327897200703478317585468, −8.56217115070958397678701035587, −7.88512387937439276689618058178, −6.9848846973708690703746375641, −4.67056332205267766088074924527, −3.57122375294470579340249670053, −2.004138956328056952640748714220, −0.82314205054305045360686249429,
2.17833489774132951969276770339, 3.831935311648663800222044575977, 5.0445742604780906933608795455, 6.41287462517534960612984152222, 7.69509562267688104084813893877, 8.723175273318825772286242686615, 9.36240721764328995709719049594, 11.00029785279877666881289523939, 11.33389416080552239419279344478, 13.69923403585927102218327002950, 14.623779279180970152049197159455, 15.26495561001823182618017304781, 16.06742784219355100718734111521, 17.15570891457015068937282968514, 18.42593401131291475860639635724, 19.26364772101384118600453348926, 19.99859814090580508601467804799, 21.65443512561266368822942213110, 22.123456525725224421124463192919, 23.65152713228251702760093570428, 24.37875507001712989109750640271, 25.544379374856381885517138556980, 26.399114607667709380390546349150, 27.02216920100942632564193532712, 27.782986841580161262738724567623