L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.104 + 0.994i)3-s + (−0.104 − 0.994i)4-s + (0.309 − 0.951i)5-s + (0.669 + 0.743i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (−0.809 + 0.587i)18-s + (0.913 − 0.406i)19-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.104 + 0.994i)3-s + (−0.104 − 0.994i)4-s + (0.309 − 0.951i)5-s + (0.669 + 0.743i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (−0.809 + 0.587i)18-s + (0.913 − 0.406i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0696 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0696 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.035438210 - 0.9656500391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035438210 - 0.9656500391i\) |
\(L(1)\) |
\(\approx\) |
\(1.209495640 - 0.6058031547i\) |
\(L(1)\) |
\(\approx\) |
\(1.209495640 - 0.6058031547i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (-0.104 - 0.994i)T \) |
| 17 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.913 + 0.406i)T \) |
| 41 | \( 1 + (-0.104 + 0.994i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.978 - 0.207i)T \) |
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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
−28.8917912959496902113905937341, −27.37289402413832324554778409765, −26.10291375929875243335792197269, −25.31646967717628773777867828943, −24.74166198578811618463964544640, −23.59097484534268352639683112449, −22.665840828657845144768519257168, −22.04098156402877798349001122561, −20.84032613010938310811849303033, −19.21338095022689389803425088678, −18.25603931057825253520009264512, −17.677893362806011369867892608469, −16.29019139315212508169755718795, −15.139654955258483588182052175572, −14.16512072136807944256061743082, −13.43488435254764592741230772228, −12.12139053172585571148548960309, −11.528625175267360170949126986669, −9.56732553103984307761027835405, −8.09691190138632694520920772131, −7.206661231059254017005541025653, −6.1115174992545477559627099743, −5.41653914289773927622637234859, −3.29039233855778535780545019036, −2.29506060886130325471774366971,
1.12959939951812032535919366595, 3.10452718485932858430581726063, 4.29402796528333613233652538521, 5.043881461501422583628053255803, 6.290935908294679047237156417031, 8.39820815676684605582812375205, 9.72855290502328867688748167577, 10.29537369826723423728813788792, 11.50937214607958666264126634830, 12.62309839376540034584558398059, 13.73615475228767550528884758470, 14.581937768276154081400778329880, 15.945764291638957715260302000935, 16.726460748374393563274312949334, 17.93738908587138614738575516908, 19.77185576517139162416697098864, 20.17673033407253321586437779925, 21.16550103624072676205015073427, 21.84866795645795947249308849630, 23.05073345450511961602110551308, 23.74832963529411521210018583199, 24.96388493211570978712980066896, 26.350819031900377855282767947872, 27.311822912461693727323750228010, 28.32038380964970466166635655132