L(s) = 1 | + (0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (−0.913 + 0.406i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (0.309 − 0.951i)18-s + (0.978 + 0.207i)19-s + (−0.104 − 0.994i)20-s + ⋯ |
L(s) = 1 | + (0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (−0.913 + 0.406i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (0.309 − 0.951i)18-s + (0.978 + 0.207i)19-s + (−0.104 − 0.994i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.338931976 + 1.248452603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.338931976 + 1.248452603i\) |
\(L(1)\) |
\(\approx\) |
\(1.215172578 + 0.5954218333i\) |
\(L(1)\) |
\(\approx\) |
\(1.215172578 + 0.5954218333i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.913 - 0.406i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.978 - 0.207i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.978 - 0.207i)T \) |
| 41 | \( 1 + (-0.669 + 0.743i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.669 + 0.743i)T \) |
| 61 | \( 1 + (0.913 - 0.406i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.913 + 0.406i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.104 - 0.994i)T \) |
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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
−21.91696087893780403639988445378, −20.68538809990827989256413192317, −19.77488646771321519737456277504, −19.25801207875040806233530049868, −18.5110326713521250636063812083, −17.772461135808699884838355938165, −16.50935434967655499478046320654, −15.94474190921165324495817060048, −14.99965679666017188492196965365, −14.20174213028475703886345740415, −13.47151838505777146744265700969, −12.54440130117952308873882125208, −11.88984955392052652723325715767, −11.36812567710682438443053802653, −10.579293930849774464498164167216, −9.7202900463396217446125831591, −8.03599958113050685561099282491, −7.398802006477003656722644935665, −6.59246000216496774559266519565, −5.74571296767639032151589591232, −4.94809687905193109438554844174, −3.82788908518135291967535116455, −3.02088424054539881697604685345, −1.91806458606253108040634956366, −0.78932904509245554336693253206,
1.04272835155901415622786375741, 2.866279404127683887460225600, 3.72769059723160000717977735385, 4.480101140019207034927945752095, 5.20440990190238729719256392440, 5.931209014151221128336624700053, 7.00448283508414131933436325171, 7.86156289631381457465109359781, 8.7680462011272626369523620889, 9.8618369804776761620737818992, 10.869038542526156556292089180654, 11.741061690986409403423043277574, 12.16106081203761994135564046157, 12.93075552093842235701509585164, 14.17684715832624735603043550909, 14.74610311364756324485054932387, 15.83649282504828396838815372286, 16.083222146854498709396007504882, 16.75366140722677546267805071300, 17.61494809084874551879301920285, 18.63533439694446041218745366285, 19.94516067807672893901527651436, 20.49450735408493629794816737729, 21.12594539481542667147071004337, 22.06570225252434112048976266202