L(s) = 1 | + (0.637 + 0.770i)3-s + (−0.968 − 0.248i)5-s + (−0.637 − 0.770i)7-s + (−0.187 + 0.982i)9-s + (0.809 + 0.587i)11-s + (−0.728 − 0.684i)13-s + (−0.425 − 0.904i)15-s + (−0.929 − 0.368i)17-s + (0.992 + 0.125i)19-s + (0.187 − 0.982i)21-s + (−0.992 + 0.125i)23-s + (0.876 + 0.481i)25-s + (−0.876 + 0.481i)27-s + (0.187 − 0.982i)29-s + (−0.187 + 0.982i)31-s + ⋯ |
L(s) = 1 | + (0.637 + 0.770i)3-s + (−0.968 − 0.248i)5-s + (−0.637 − 0.770i)7-s + (−0.187 + 0.982i)9-s + (0.809 + 0.587i)11-s + (−0.728 − 0.684i)13-s + (−0.425 − 0.904i)15-s + (−0.929 − 0.368i)17-s + (0.992 + 0.125i)19-s + (0.187 − 0.982i)21-s + (−0.992 + 0.125i)23-s + (0.876 + 0.481i)25-s + (−0.876 + 0.481i)27-s + (0.187 − 0.982i)29-s + (−0.187 + 0.982i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075329856 + 0.5030389232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075329856 + 0.5030389232i\) |
\(L(1)\) |
\(\approx\) |
\(0.9178204276 + 0.1575176521i\) |
\(L(1)\) |
\(\approx\) |
\(0.9178204276 + 0.1575176521i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (0.637 + 0.770i)T \) |
| 5 | \( 1 + (-0.968 - 0.248i)T \) |
| 7 | \( 1 + (-0.637 - 0.770i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.728 - 0.684i)T \) |
| 17 | \( 1 + (-0.929 - 0.368i)T \) |
| 19 | \( 1 + (0.992 + 0.125i)T \) |
| 23 | \( 1 + (-0.992 + 0.125i)T \) |
| 29 | \( 1 + (0.187 - 0.982i)T \) |
| 31 | \( 1 + (-0.187 + 0.982i)T \) |
| 37 | \( 1 + (-0.876 - 0.481i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (-0.968 + 0.248i)T \) |
| 47 | \( 1 + (0.968 + 0.248i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.535 + 0.844i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + (0.187 - 0.982i)T \) |
| 71 | \( 1 + (-0.425 + 0.904i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.992 + 0.125i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.968 + 0.248i)T \) |
| 97 | \( 1 + (-0.637 + 0.770i)T \) |
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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
−17.86291627913592718972567789761, −16.85378879604891548628012636784, −16.31811765905680256077816798838, −15.399350751030915501593578840859, −15.14705431806729590491347145153, −14.24093299277958780780398549179, −13.82612842580566212356985764855, −12.95951641336376713251815019042, −12.27159130918066715203190004311, −11.746989378260032789864947307746, −11.43302331128103148424311083464, −10.20517397902574949427994562762, −9.45019660290248821522691142892, −8.731501208275120041177724069289, −8.43708054536286323732247935908, −7.45642609851147924110241124478, −6.86435827047408844461994259311, −6.417511386450185731420692121459, −5.54450055968077237580346708427, −4.44909270834976179881027451287, −3.67011116725003081865892939867, −3.15724681440052962195924514428, −2.35461812066347699945745157589, −1.60990194079959138965733751458, −0.43821638119936845673354548991,
0.59271241434101131266413436884, 1.81698494129220951122101780733, 2.76801729199124004200436575746, 3.5913916410652118200582062250, 3.92496236030886882977579189765, 4.71169446506568301965279453628, 5.27862380547450614289445366453, 6.498133058988741243656220998187, 7.319631986910254610261070975199, 7.619687623717669219018999040759, 8.57471299060629881426035511566, 9.11769686817757450817650269866, 9.98902031004806550773270493543, 10.2153466051653735242080868195, 11.18933009168801900639571503140, 11.888917260723971188930078636304, 12.468628465810330983347680912082, 13.35189755190952118417221450136, 13.96282102282891904271475245490, 14.55451092448916817996695196371, 15.42516910455297935560564106453, 15.70581830062640023742068709287, 16.32892316060403705241529085969, 17.05012713946983401156662520962, 17.6028285747000397808648049715