L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.913 − 0.406i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.809 + 0.587i)20-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.913 − 0.406i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.809 + 0.587i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1226950679 - 0.3511286066i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1226950679 - 0.3511286066i\) |
\(L(1)\) |
\(\approx\) |
\(0.3272553678 - 0.4587057206i\) |
\(L(1)\) |
\(\approx\) |
\(0.3272553678 - 0.4587057206i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.104 - 0.994i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.913 - 0.406i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.104 - 0.994i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−32.01767021116261060295324537358, −31.29278252352979224035553322082, −29.86967130395994092091347907166, −28.25331312530179917013721858747, −27.42433912499378830723868560875, −26.602057473281608858841176336555, −25.7052950695354949286363112538, −23.8690807796881950269936524642, −23.55973495870992065529327344383, −22.21594039936161846536311365967, −21.62293101319792692831996549831, −19.735192270341948431214124200327, −18.44819395441430824906792818551, −17.18138161006737324924908036010, −16.35774171131141013908194527800, −15.20859186420637194141544998292, −14.624661599788439498836812524922, −12.73944595128524385175767609207, −11.49487317795740836713636160868, −10.12720929337523328390176863726, −8.75318597444822468941823126724, −7.31558767720746658554243146486, −6.14170461017547026166533397239, −4.676868133353352725910319757672, −3.70030512299410343531230709086,
0.43848158120631132866764026584, 2.36816056014424033241386735469, 4.24701597376632252383815978859, 5.470744227307850620940774120219, 7.37877744830931553683305296653, 8.61695049597304819532942938006, 10.33626417667269482302335223532, 11.51262593668324876113620344397, 12.32240055879339962641219867029, 13.17625675491781081396558123225, 14.6960027932465767293734673534, 16.40517879352628457890510350491, 17.59594938252360160092776104101, 18.71578287011481195209877691419, 19.59157998960460313997574232703, 20.54924402116639417415433718517, 22.10155790889310498336140989440, 22.94970454446800460615169487392, 23.81539020702033754431881718711, 24.90544663947560325893002085091, 26.84853658552083510482659842172, 27.70135740049758784147028068908, 28.56919227828043923484176028198, 29.59913052311906155662101743854, 30.41744985862039923895407922863