L(s) = 1 | + (−0.984 − 0.173i)2-s + (−0.984 + 0.173i)3-s + (0.939 + 0.342i)4-s + 6-s + (−0.642 − 0.766i)7-s + (−0.866 − 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.984 − 0.173i)12-s + (0.342 − 0.939i)13-s + (0.5 + 0.866i)14-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)18-s + (−0.173 − 0.984i)19-s + ⋯ |
L(s) = 1 | + (−0.984 − 0.173i)2-s + (−0.984 + 0.173i)3-s + (0.939 + 0.342i)4-s + 6-s + (−0.642 − 0.766i)7-s + (−0.866 − 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.984 − 0.173i)12-s + (0.342 − 0.939i)13-s + (0.5 + 0.866i)14-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)18-s + (−0.173 − 0.984i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001735743167 + 0.007670715222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.001735743167 + 0.007670715222i\) |
\(L(1)\) |
\(\approx\) |
\(0.4231703395 - 0.04672981749i\) |
\(L(1)\) |
\(\approx\) |
\(0.4231703395 - 0.04672981749i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.984 + 0.173i)T \) |
| 7 | \( 1 + (-0.642 - 0.766i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.342 - 0.939i)T \) |
| 17 | \( 1 + (0.342 + 0.939i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 - 0.939i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 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
−26.810498468125416415181308210978, −25.58060556921238568687776410370, −24.796453115330976053871196437062, −23.804265180960364446115027035460, −22.94648986904706244986550339910, −21.63680128837789904449938188072, −20.92359700702082316561654813764, −19.23722955895563213316113081494, −18.736980950462664414359235963418, −17.99809083521955467688889607508, −16.61287072266506706151060755748, −16.28598587030725011334882376836, −15.275485397170995451131216019698, −13.66151807930597948417255759765, −12.23899757331251985335092921627, −11.53844695684383995662427864255, −10.48691268489626808807302954736, −9.467747871213493831284333866842, −8.36886581506738149638597921681, −7.01293117955296083432598351819, −6.15696572249820371455330601659, −5.2126442777857824924902691128, −3.068168924416104674900640093568, −1.46644554687432738741412364212, −0.00509826044300079497125422285,
1.16893889787005936712124050699, 3.01354609047521244024142600795, 4.54233310094947742157468321364, 6.12147837255287774320675893549, 6.9812885013756711272189025153, 8.07965591588111001653692849371, 9.65696087439285525925721307860, 10.36202399137904207507718075906, 11.0758917794480458183866645427, 12.43133015144694735305394983739, 13.09522996295900357375958658268, 15.235247520446118979691118388905, 15.83643984977893718603020715682, 17.09756208954484664295605152008, 17.4299978971642583275127799765, 18.5444003486117098375805532742, 19.58286798222501585831496069951, 20.57321808465115008209326075283, 21.47854085417085280546173233560, 22.76384243423272594679475644576, 23.43193916642980492743430490668, 24.63498956133545339527740176185, 25.80620797096109427439005472254, 26.50656933915120544332841977861, 27.53112003744494276694065221289