L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 − 0.342i)5-s + 7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + (−0.5 − 0.866i)20-s + (0.766 − 0.642i)22-s + (0.766 + 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 − 0.342i)5-s + 7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + (−0.5 − 0.866i)20-s + (0.766 − 0.642i)22-s + (0.766 + 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6331668679 - 0.2803628902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6331668679 - 0.2803628902i\) |
\(L(1)\) |
\(\approx\) |
\(0.6761075882 - 0.1599830948i\) |
\(L(1)\) |
\(\approx\) |
\(0.6761075882 - 0.1599830948i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.173 - 0.984i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.173 - 0.984i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−27.45176872202182149425119021208, −26.79266871005804332347443494338, −26.048296816813311023426767849860, −24.77038596758858813438658989985, −23.79169167133175323142164568019, −23.425474408998372774573520032125, −21.61451558174351388989209712405, −20.746040767140101757555452735366, −19.55736148086647780402409434425, −18.786132378561752041784655609395, −18.060763506282666787396382060385, −16.72144626546982072289426481029, −16.06456721537340568657782121721, −14.84291081923440843816379243317, −14.23720137578068177376891570502, −12.30320234443434368476787225517, −11.11280151882827459655208202900, −10.75413513014287444163332173855, −9.03675480341086876393044689020, −8.15651001939093460642310181983, −7.36356637393620064415912741058, −6.08578064999205083519480372960, −4.645899165049847202684574960107, −2.97114031567055852879169135583, −1.27798719603702888769069336630,
0.94522201789676153551429068571, 2.54882907363200954448686492762, 4.01233067882802289218159102634, 5.39445497217030236808354230846, 7.47192193346895414984369001498, 7.76621805233389662547700080552, 8.962819751188664922793546713035, 10.21381780770876783751357362857, 11.26400323056048485848929294869, 12.043272080042759983513745716229, 13.08479909034284741213857044671, 14.94684946449684351661674570671, 15.585686698796235638154242482596, 16.76922360781144319443533242886, 17.73570991039910364241359612449, 18.54956524026048695955996559757, 19.64335853012185047740046842143, 20.59188142061195743213967574033, 21.020897816642277057589498743879, 22.65780953097515984569363777779, 23.65556296557538034984023522917, 24.72904364119934838992859869444, 25.54700985195082065692200260363, 26.77561627630916439265425048153, 27.59773228874359816526152130258