L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)10-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 20-s + (−0.939 + 0.342i)22-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)10-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 20-s + (−0.939 + 0.342i)22-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1095044611 - 0.4430338957i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1095044611 - 0.4430338957i\) |
\(L(1)\) |
\(\approx\) |
\(0.5269534825 - 0.2153997467i\) |
\(L(1)\) |
\(\approx\) |
\(0.5269534825 - 0.2153997467i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.766 + 0.642i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (-0.173 - 0.984i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.939 + 0.342i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.766 + 0.642i)T \) |
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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
−33.16027407555831098837076938243, −32.19747121785004769244435495824, −31.23380112190428637465578719867, −29.2909453847042799453240524844, −28.349708152662496045847123114111, −27.66771384419409890629153858304, −26.29537224976903276901355696846, −25.10649725740110295995782997353, −24.45049720385606447035761003130, −23.19847558610379229459833955286, −21.724272000965295816309728551183, −20.01934332890814020309280654554, −19.36698403501722680963841741743, −17.83866014513080399483949204999, −16.860334397040323183819045380193, −15.728666569528201419100953513053, −14.75721394270663008155597160572, −12.912997588085638803202788068100, −11.653777971213873554829757196002, −9.68729486678892529765002192469, −8.99891870644067934006034512742, −7.56394257509062582759670801054, −6.05295160870750965616934356035, −4.65494985912938252198334564267, −1.85928442957834021961114120954,
0.31032801689692085046430111095, 2.636696046128937492552391647707, 3.9219705746898476961435825694, 6.63511564268361791620560801616, 7.67813679513381251277454824662, 9.37319121821096221038210223218, 10.55255962001348342080489429236, 11.45151339722021563482407412145, 13.01484166689710653408882182677, 14.39453065089934686259602307575, 16.114508525081480409972112966139, 17.20315828308083984296408877971, 18.454562424893920846373133233533, 19.45362426073574597009121020397, 20.334700950711180207522243271677, 21.99343378115916539365552502087, 22.60111156483352625615865907061, 24.40293059136121626106695570184, 25.87336330945600888532474087098, 26.74302910357759349347507729259, 27.38846001025252461652233716011, 29.14528161978252679825704014864, 29.69439203214172308582266913931, 30.67582810973950004143106875585, 32.01300507490969865060327138366