L(s) = 1 | + (0.923 + 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.382 + 0.923i)13-s − i·15-s − i·17-s + (0.382 − 0.923i)19-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + 31-s + 33-s + (−0.382 − 0.923i)37-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.382 + 0.923i)13-s − i·15-s − i·17-s + (0.382 − 0.923i)19-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + 31-s + 33-s + (−0.382 − 0.923i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.460237222 - 1.163605546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.460237222 - 1.163605546i\) |
\(L(1)\) |
\(\approx\) |
\(1.465554581 - 0.1916271039i\) |
\(L(1)\) |
\(\approx\) |
\(1.465554581 - 0.1916271039i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.923 + 0.382i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 11 | \( 1 + (0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.382 - 0.923i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
| 29 | \( 1 + (0.923 + 0.382i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.382 - 0.923i)T \) |
| 41 | \( 1 + (0.707 + 0.707i)T \) |
| 43 | \( 1 + (-0.923 + 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.923 - 0.382i)T \) |
| 59 | \( 1 + (-0.382 - 0.923i)T \) |
| 61 | \( 1 + (0.923 + 0.382i)T \) |
| 67 | \( 1 + (-0.923 - 0.382i)T \) |
| 71 | \( 1 + (0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (0.382 - 0.923i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
| 97 | \( 1 + T \) |
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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
−24.00209856665887500192043450106, −23.02142476268424052610175220395, −22.27803075500873152762277896198, −21.33634613369611641734429903273, −20.24105996432646963175614674991, −19.56682303355015292673022045903, −18.98703387358744458167681714384, −17.96451337977873223844678975950, −17.27216883176757094339049818329, −15.73508915942301075415405935720, −15.0890682118223666794074169795, −14.367826857366634673799958273087, −13.63374245881936680617524867158, −12.394044734835477676904110678776, −11.806235863997117261113799529703, −10.334540068089939522815552269302, −9.80827470624800255505598533791, −8.42972533946280828876001694874, −7.776799314003492742213434678931, −6.85622338819227692787688537943, −5.95382422631339436261070885269, −4.18556645314043620689514358578, −3.43124904342540537948355395634, −2.42457217225829214429236628658, −1.21634671098759452439246462943,
0.69788098751995420033116329901, 2.01988727010445815988628167976, 3.2580262827992762114360968198, 4.36064082321187898289048363726, 4.92942291038651190037501566417, 6.5797949448848740066309413813, 7.59592275392707582457432785554, 8.68984844928777632471326815289, 9.13998945537040479077306781671, 10.056719958306537302747350995998, 11.48730960346634326454035043851, 12.14392805467011735863600267521, 13.38640131796319552624738516508, 14.035652001954074424836596279948, 14.89990682589818117352605502175, 16.145732836194491463392800650732, 16.30035123576500212242395480650, 17.55915725547723421498629350857, 18.78471634210296718830170857461, 19.73850690631916205510685288826, 20.00559929097875308906870220202, 21.119314204139453239319476476191, 21.71252707883428382269467179095, 22.75018541536192159265283785269, 23.97041173022086527427544885350