L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (−0.173 − 0.984i)4-s + 6-s + (0.342 − 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 + 0.766i)12-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)18-s + (−0.766 + 0.642i)19-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (−0.173 − 0.984i)4-s + 6-s + (0.342 − 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 + 0.766i)12-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)18-s + (−0.766 + 0.642i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9938519812 - 0.1306128015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9938519812 - 0.1306128015i\) |
\(L(1)\) |
\(\approx\) |
\(0.7033040804 + 0.005700290582i\) |
\(L(1)\) |
\(\approx\) |
\(0.7033040804 + 0.005700290582i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 7 | \( 1 + (0.342 - 0.939i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (0.984 + 0.173i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.342 + 0.939i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.984 - 0.173i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−27.345886673610022238671164817102, −26.26325517067215954106983572922, −25.53503541907878011827907017866, −24.05712583455639235498325855520, −22.93554894466139926697903418217, −21.819311953343195590777781257, −21.29887547916249169165374122944, −20.57199730999375215619175235083, −19.09827564153929219256713227082, −18.33356893290350562784350475866, −17.47892272993911941163867706885, −16.286527188650887867438631024845, −15.730691544270215141372536165384, −14.21509257560237693336207701368, −12.75798011461011393887069664356, −11.77748001645737063471571033238, −11.01254594401028992112443243501, −10.136115868671544735132745452826, −8.91601967400464353988621798908, −8.23991523062702801124781942264, −6.36033175702742859430346851094, −5.16811010338834436847843625021, −3.83751256754366245288345275477, −2.63386270490628954054125821699, −0.85085971077589220691615886615,
0.72498089801185228273341194381, 1.843765672351886560099416568638, 4.30881434643379244637645962001, 5.59870717406762052642444308724, 6.54399833165449891599570859224, 7.662096172686239094467269836585, 8.20317229245109758665481444975, 10.0564566949450459766491092764, 10.65773891420569162240747655129, 11.96783708385252835003641854848, 13.31978579774038752657462266685, 14.09568218032136888942923150400, 15.392980276780249222357068374966, 16.50675123241260710448641534473, 17.28094146518205075013578213664, 18.0345586496004873498543175555, 18.881587437968247493515066633731, 19.919005003182374204786518055640, 21.019601254526707872370021458678, 22.84781501125016261116686882911, 23.31402215759144993060659090818, 23.94131191489545047224523080105, 25.18883516573071852921091483124, 25.7567913865039971862039091315, 26.94455382171495548914875168716