L(s) = 1 | + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.173 − 0.984i)5-s + (−0.374 − 0.927i)7-s + (0.669 − 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.615 − 0.788i)11-s + (−0.961 − 0.275i)13-s + (−0.374 + 0.927i)14-s + (−0.997 + 0.0697i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.990 + 0.139i)20-s + (−0.990 + 0.139i)22-s + (0.615 + 0.788i)23-s + ⋯ |
L(s) = 1 | + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.173 − 0.984i)5-s + (−0.374 − 0.927i)7-s + (0.669 − 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.615 − 0.788i)11-s + (−0.961 − 0.275i)13-s + (−0.374 + 0.927i)14-s + (−0.997 + 0.0697i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.990 + 0.139i)20-s + (−0.990 + 0.139i)22-s + (0.615 + 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9127852917 - 0.1982906467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9127852917 - 0.1982906467i\) |
\(L(1)\) |
\(\approx\) |
\(0.6256428429 - 0.3105027594i\) |
\(L(1)\) |
\(\approx\) |
\(0.6256428429 - 0.3105027594i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.719 - 0.694i)T \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.374 - 0.927i)T \) |
| 11 | \( 1 + (0.615 - 0.788i)T \) |
| 13 | \( 1 + (-0.961 - 0.275i)T \) |
| 17 | \( 1 + (-0.669 + 0.743i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.615 + 0.788i)T \) |
| 29 | \( 1 + (0.719 + 0.694i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.559 - 0.829i)T \) |
| 43 | \( 1 + (0.719 + 0.694i)T \) |
| 47 | \( 1 + (0.559 + 0.829i)T \) |
| 53 | \( 1 + (-0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.961 + 0.275i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.669 - 0.743i)T \) |
| 79 | \( 1 + (-0.0348 + 0.999i)T \) |
| 83 | \( 1 + (0.241 - 0.970i)T \) |
| 89 | \( 1 + (0.978 - 0.207i)T \) |
| 97 | \( 1 + (-0.374 - 0.927i)T \) |
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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
−22.28705492422046925913020920330, −21.32610201802014948611578141584, −20.06026541609208918583508319858, −19.28691336960042006166133275063, −18.83422738867358614967544658246, −17.7970551689939780962993577694, −17.481999905583254651359463754473, −16.366134826929658157745384823946, −15.47539609208785858519979038245, −14.84650148293476787594746242025, −14.365171370171728399724266130391, −13.19564367094769560603304482299, −12.0469822804074621652382383822, −11.21593527946283843782455647366, −10.26679412717502653319740349452, −9.4546097963070594105874766205, −8.94024318490668150162396585138, −7.71245141606199084972447534428, −6.73081961875051226714430633332, −6.50831247615155649769782016502, −5.26561867922501254151070627026, −4.31928467637866487930908991016, −2.53736629593222888778502359650, −2.18701209230255622777460189518, −0.36356604722597171881635392760,
0.76193710323805916570802821997, 1.50171274194633610047392936945, 2.798638136107351646408791916843, 3.916162552892047512737136810646, 4.55376183563575418378960007764, 5.97339795744297341363479616596, 7.0313951278665429477053290477, 7.98539207185494655942008326495, 8.77382120802555408198654783499, 9.50705033006752433865991079080, 10.34990115284037494550660741965, 11.097013196612667783972985129463, 12.130311225207121481877280265288, 12.84253717187710226360414462381, 13.441511019002290372852696555505, 14.47288155496266490175295136207, 15.84587876809435228961838313693, 16.55959070219647075960318657739, 17.290537036137534516253120966389, 17.52757943324577319072981289192, 19.11458121430736373496964990585, 19.475613135167017109921574974363, 20.155135862683422424761795684919, 20.93836313438711709538502091414, 21.71140502915828043238143058791