L(s) = 1 | + (0.934 − 0.356i)2-s + (0.962 + 0.269i)3-s + (0.746 − 0.665i)4-s + (0.803 + 0.595i)5-s + (0.995 − 0.0909i)6-s + (0.291 − 0.956i)7-s + (0.460 − 0.887i)8-s + (0.854 + 0.519i)9-s + (0.962 + 0.269i)10-s + (0.854 + 0.519i)11-s + (0.898 − 0.439i)12-s + (−0.877 − 0.480i)13-s + (−0.0682 − 0.997i)14-s + (0.613 + 0.789i)15-s + (0.113 − 0.993i)16-s + (−0.990 − 0.136i)17-s + ⋯ |
L(s) = 1 | + (0.934 − 0.356i)2-s + (0.962 + 0.269i)3-s + (0.746 − 0.665i)4-s + (0.803 + 0.595i)5-s + (0.995 − 0.0909i)6-s + (0.291 − 0.956i)7-s + (0.460 − 0.887i)8-s + (0.854 + 0.519i)9-s + (0.962 + 0.269i)10-s + (0.854 + 0.519i)11-s + (0.898 − 0.439i)12-s + (−0.877 − 0.480i)13-s + (−0.0682 − 0.997i)14-s + (0.613 + 0.789i)15-s + (0.113 − 0.993i)16-s + (−0.990 − 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.418393133 - 0.9276320672i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.418393133 - 0.9276320672i\) |
\(L(1)\) |
\(\approx\) |
\(2.788761677 - 0.4209347724i\) |
\(L(1)\) |
\(\approx\) |
\(2.788761677 - 0.4209347724i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.934 - 0.356i)T \) |
| 3 | \( 1 + (0.962 + 0.269i)T \) |
| 5 | \( 1 + (0.803 + 0.595i)T \) |
| 7 | \( 1 + (0.291 - 0.956i)T \) |
| 11 | \( 1 + (0.854 + 0.519i)T \) |
| 13 | \( 1 + (-0.877 - 0.480i)T \) |
| 17 | \( 1 + (-0.990 - 0.136i)T \) |
| 19 | \( 1 + (-0.715 + 0.699i)T \) |
| 23 | \( 1 + (0.203 + 0.979i)T \) |
| 29 | \( 1 + (0.854 - 0.519i)T \) |
| 31 | \( 1 + (-0.998 + 0.0455i)T \) |
| 37 | \( 1 + (-0.877 - 0.480i)T \) |
| 41 | \( 1 + (-0.576 - 0.816i)T \) |
| 43 | \( 1 + (0.377 - 0.926i)T \) |
| 47 | \( 1 + (0.613 + 0.789i)T \) |
| 53 | \( 1 + (-0.949 - 0.313i)T \) |
| 59 | \( 1 + (-0.949 + 0.313i)T \) |
| 61 | \( 1 + (-0.419 + 0.907i)T \) |
| 67 | \( 1 + (0.460 + 0.887i)T \) |
| 71 | \( 1 + (-0.775 + 0.631i)T \) |
| 73 | \( 1 + (-0.949 + 0.313i)T \) |
| 79 | \( 1 + (-0.949 + 0.313i)T \) |
| 83 | \( 1 + (0.995 - 0.0909i)T \) |
| 89 | \( 1 + (0.538 - 0.842i)T \) |
| 97 | \( 1 + T \) |
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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
−21.76341273180541745626679446721, −21.29534634087840212845742419368, −20.27327186242013244931353418865, −19.75324723277122410357780121428, −18.76868866232525127766126267415, −17.714927964770243507966238554469, −16.99900577468062877408151834353, −16.08949787923045044318849960089, −15.18200298913645864579411004978, −14.520153482584268949653607537876, −13.97533049515818686190511466381, −13.080769590005649258258356387218, −12.49725199997352039006603522539, −11.762748281234020713672719297712, −10.58383051685917922386289950409, −9.04736636479060228925659443672, −8.96046834009267488977342461786, −7.96022268194910749070802296185, −6.620240341111292741976767685691, −6.34988637898370610292306443637, −4.93279393785048104827038214841, −4.481988514457939441681940731967, −3.13520147084941713889026866623, −2.28279404150595115830241518246, −1.66576445303412998145226203885,
1.559784027279162512794213694200, 2.15043068608434095247539398191, 3.184469072964223385092634195217, 4.03249314949358552273693283748, 4.738277627409323768142825149491, 5.87977308504812392686094884321, 7.101982681741240313316086677319, 7.30584488736841113838209604614, 8.88037919542996731653759252268, 9.85128645373258782900740210686, 10.34872932478329370265082596390, 11.099299087540442032056482115204, 12.32182141718231202134492832348, 13.16203591075627325845973553971, 13.89465407462660567586026774445, 14.37993649796477711093108079027, 14.99440244104858873830177722375, 15.781376472909979291079202783503, 17.055374534291296393967197800172, 17.6385750501159802063126418316, 18.98656301586187216933020526710, 19.57670124081641155834965554881, 20.3153584009879517637149011273, 20.85347129832329976390590812662, 21.76412757379775598235787470575