L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (0.809 + 0.587i)5-s + (−0.913 + 0.406i)6-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.309 + 0.951i)18-s + (0.978 + 0.207i)19-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (0.809 + 0.587i)5-s + (−0.913 + 0.406i)6-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.309 + 0.951i)18-s + (0.978 + 0.207i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7734560299 - 0.5816469540i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7734560299 - 0.5816469540i\) |
\(L(1)\) |
\(\approx\) |
\(0.8566124220 - 0.3802808825i\) |
\(L(1)\) |
\(\approx\) |
\(0.8566124220 - 0.3802808825i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.669 - 0.743i)T \) |
| 17 | \( 1 + (0.913 - 0.406i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.978 + 0.207i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.978 - 0.207i)T \) |
| 41 | \( 1 + (-0.669 + 0.743i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.913 - 0.406i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.913 + 0.406i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.104 + 0.994i)T \) |
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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
−28.326800620585886521516716793156, −27.533019695351646650588515897542, −26.35897160806703960682793757506, −25.55061341129095114913579710357, −25.08578500991105413631848297158, −23.99185838997360314592082992652, −22.33994348863879594788154626475, −21.32665856891749372407445180131, −20.433618138330379915029370863119, −19.4839200116851453384915322545, −18.51078059168226174866721266041, −17.27128194892023173836538232983, −16.3103952270835253668741983870, −15.598212854025226017827357798047, −14.48867896943153835287535234367, −13.38508122669743531172981761359, −11.84887073741313251886225784865, −10.23644206644807177445989013142, −9.57522394279348078000105648287, −8.84486887966424626274515825858, −7.720777294466040814311237294284, −6.02854204499571178529001523091, −5.16103319580872286889119253764, −3.12643168507196764719539970518, −1.77769552624470232221383425014,
1.19913635269388515269982447095, 2.598066316017325963060774395096, 3.51469234854974958917998562155, 6.18283408187068803060509364837, 7.106655172015880575273878621751, 8.02724172663363911994592775553, 9.540451173376345981818279706135, 10.003363513429710340757859645332, 11.45577946539699098453061454753, 12.73466795611334620836264244560, 13.62706217163996815420950749010, 14.68940744830668035225184489330, 16.27088233329452878803074163357, 17.27070026870268906934955851765, 18.3761612773255613553893303363, 18.84792395165605756368423148285, 20.08990467022694710145169322329, 20.671187957687280256701463991561, 21.97048682732805404619221280912, 23.1512812265979641793970259605, 24.6525715169998460132094442601, 25.32409232252304007658768973135, 26.348658798950029422489673565317, 26.61413622912881359790497434504, 28.30151774496910144211770315450