L(s) = 1 | + (0.646 − 0.762i)2-s + (−0.163 − 0.986i)4-s + (0.420 − 0.907i)5-s + (−0.858 − 0.512i)8-s + (−0.420 − 0.907i)10-s + (−0.791 + 0.611i)11-s + (−0.983 − 0.178i)13-s + (−0.946 + 0.323i)16-s + (−0.772 − 0.635i)17-s + (0.104 + 0.994i)19-s + (−0.963 − 0.266i)20-s + (−0.0448 + 0.998i)22-s + (−0.251 + 0.967i)23-s + (−0.646 − 0.762i)25-s + (−0.772 + 0.635i)26-s + ⋯ |
L(s) = 1 | + (0.646 − 0.762i)2-s + (−0.163 − 0.986i)4-s + (0.420 − 0.907i)5-s + (−0.858 − 0.512i)8-s + (−0.420 − 0.907i)10-s + (−0.791 + 0.611i)11-s + (−0.983 − 0.178i)13-s + (−0.946 + 0.323i)16-s + (−0.772 − 0.635i)17-s + (0.104 + 0.994i)19-s + (−0.963 − 0.266i)20-s + (−0.0448 + 0.998i)22-s + (−0.251 + 0.967i)23-s + (−0.646 − 0.762i)25-s + (−0.772 + 0.635i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.111873826 + 0.02273131854i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.111873826 + 0.02273131854i\) |
\(L(1)\) |
\(\approx\) |
\(0.9859724944 - 0.6207186217i\) |
\(L(1)\) |
\(\approx\) |
\(0.9859724944 - 0.6207186217i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.646 - 0.762i)T \) |
| 5 | \( 1 + (0.420 - 0.907i)T \) |
| 11 | \( 1 + (-0.791 + 0.611i)T \) |
| 13 | \( 1 + (-0.983 - 0.178i)T \) |
| 17 | \( 1 + (-0.772 - 0.635i)T \) |
| 19 | \( 1 + (0.104 + 0.994i)T \) |
| 23 | \( 1 + (-0.251 + 0.967i)T \) |
| 29 | \( 1 + (0.550 - 0.834i)T \) |
| 31 | \( 1 + (0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.447 - 0.894i)T \) |
| 43 | \( 1 + (-0.393 + 0.919i)T \) |
| 47 | \( 1 + (-0.646 + 0.762i)T \) |
| 53 | \( 1 + (0.163 + 0.986i)T \) |
| 59 | \( 1 + (0.992 - 0.119i)T \) |
| 61 | \( 1 + (-0.712 + 0.701i)T \) |
| 67 | \( 1 + (0.669 - 0.743i)T \) |
| 71 | \( 1 + (0.550 + 0.834i)T \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.337 + 0.941i)T \) |
| 97 | \( 1 + (-0.309 - 0.951i)T \) |
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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
−17.58849726415961403331831980339, −17.087166572739901304286134495241, −16.30475922117336377837150727638, −15.551240385389405344337600864494, −15.1422895622572853310193424345, −14.419832703963329555953702020185, −13.880461265712693714698549312728, −13.28513368907215512331897020480, −12.69241009056211522938471222245, −11.83125264089257042304067135848, −11.19931802628620175561670465796, −10.40723235860759300242560253225, −9.84476289657041328923399799239, −8.70964101396203640208691907306, −8.362742012625698143116854321473, −7.414101391226702035621958206789, −6.63919955334651508020114154103, −6.54089675224422265942896632670, −5.40173888240648711468017308809, −4.968430481227081600709157287002, −4.12816345190364335640702362911, −3.149642474951574077286949231043, −2.68544901870386204129269858769, −1.98273003811991796314988381451, −0.2383086581552500045575919386,
0.85905527552764765872189898319, 1.79944573289279536707481895396, 2.37491193547132626351389082360, 3.086390503040325178833973486316, 4.27605911355652493059236190876, 4.5911069454745107810498365093, 5.381941182185870148517924265668, 5.842794642379981096077765123560, 6.79776556521916345729266944923, 7.71091400851265669272141724001, 8.40136922827026343248861419316, 9.468399478409729629222672284881, 9.73581248394243002347513921336, 10.318223526729943018535563976606, 11.2467375980513303974443464734, 11.98431971443958758678346235275, 12.43385736021448393083251882192, 13.03312224382367150116891542899, 13.68133063754203784535753351779, 14.16004600701510744884300546642, 15.08116676449093754906112379585, 15.63977046837554665870717744492, 16.22637607208397014255822904565, 17.22773254082731528597358292355, 17.76453553013020858211818944503