L(s) = 1 | + (0.309 + 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (0.913 + 0.406i)11-s + (−0.978 − 0.207i)12-s + (−0.978 + 0.207i)13-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.913 − 0.406i)17-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (0.913 + 0.406i)11-s + (−0.978 − 0.207i)12-s + (−0.978 + 0.207i)13-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.913 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6756657285 + 0.5776039891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6756657285 + 0.5776039891i\) |
\(L(1)\) |
\(\approx\) |
\(0.9466239498 + 0.5760649265i\) |
\(L(1)\) |
\(\approx\) |
\(0.9466239498 + 0.5760649265i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.669 + 0.743i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.104 - 0.994i)T \) |
| 11 | \( 1 + (0.913 + 0.406i)T \) |
| 13 | \( 1 + (-0.978 + 0.207i)T \) |
| 17 | \( 1 + (0.913 - 0.406i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.809 - 0.587i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (-0.978 - 0.207i)T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.104 + 0.994i)T \) |
| 59 | \( 1 + (0.669 + 0.743i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.104 + 0.994i)T \) |
| 73 | \( 1 + (0.913 + 0.406i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.669 - 0.743i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−36.8788987498810675440902823508, −35.487149214720580211009000838153, −34.38146975391731306504995047522, −32.14609240236512279102157033300, −31.51291275725207496643567390610, −30.21068024239181020196484360145, −29.67980776977912572757483063903, −27.965236292829883063114274070749, −26.74830677672859430859784633550, −25.21167947532579463280691103415, −23.810159784990289364464943417007, −22.47095610068285378340165263376, −21.32950820760234141486076887864, −19.44602046661103112065469116465, −19.18371877292355200069124583341, −17.818103654721401176372607157623, −15.00667848942998652788511647041, −14.25816822969707008442666371255, −12.52189038143484739139124825745, −11.639210467058220961159366883374, −9.72774467302622911284971283302, −8.17947387216985581853866804797, −6.17401232580066548652735631454, −3.605754751216236908346997776853, −2.26173985232878919099539228391,
3.85538862000002535765345155152, 4.82776541557434221420100060105, 7.21953062969992089326348503750, 8.55667006339715712275534777112, 9.86820992216816489589052736577, 12.32516288098414629694363362447, 13.89869062915112927556684950320, 14.94212877111189000353165290445, 16.40171818278226499151970449319, 17.04427187714010050963025633500, 19.467469400239010781445423815238, 20.598321831596921559435209529356, 22.05689469312930206108918034226, 23.40542431989301985708043711682, 24.64578807537855935878148189073, 25.79875751473201848093177300463, 27.05549716005993362074279443405, 27.771825679967877523368566115733, 30.09998200434084300734741158210, 31.48696252383850320706497148183, 32.339360498734148490628773438617, 33.10321064241311408392549515490, 34.46477938487047216566875458989, 36.17193781837376906572514859064, 36.46841241280649329799470010070