L(s) = 1 | + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.766 − 0.642i)5-s + (−0.615 − 0.788i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.374 + 0.927i)11-s + (−0.719 − 0.694i)13-s + (0.990 + 0.139i)14-s + (−0.997 − 0.0697i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.615 − 0.788i)20-s + (−0.374 − 0.927i)22-s + (−0.615 + 0.788i)23-s + ⋯ |
L(s) = 1 | + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.766 − 0.642i)5-s + (−0.615 − 0.788i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.374 + 0.927i)11-s + (−0.719 − 0.694i)13-s + (0.990 + 0.139i)14-s + (−0.997 − 0.0697i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.615 − 0.788i)20-s + (−0.374 − 0.927i)22-s + (−0.615 + 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08468297636 - 0.2645860402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08468297636 - 0.2645860402i\) |
\(L(1)\) |
\(\approx\) |
\(0.6064247115 + 0.005581243438i\) |
\(L(1)\) |
\(\approx\) |
\(0.6064247115 + 0.005581243438i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.719 + 0.694i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.615 - 0.788i)T \) |
| 11 | \( 1 + (-0.374 + 0.927i)T \) |
| 13 | \( 1 + (-0.719 - 0.694i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.615 + 0.788i)T \) |
| 29 | \( 1 + (-0.719 + 0.694i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.438 - 0.898i)T \) |
| 43 | \( 1 + (-0.241 - 0.970i)T \) |
| 47 | \( 1 + (0.559 - 0.829i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.241 + 0.970i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.978 + 0.207i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.882 - 0.469i)T \) |
| 83 | \( 1 + (-0.719 + 0.694i)T \) |
| 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
−22.09128368802605022099570169591, −21.76079315526871384227290103595, −20.90601379390386903709503349907, −19.9787152504714996345346735734, −19.029009031936007770029590658877, −18.584357673080360831057166099690, −17.95823243767177786963435848366, −16.96112581836003196495425265086, −16.25172765819438168020904444125, −15.3924700017747450571254440998, −14.16463258957428913590368074023, −13.46061510263542663536766179831, −12.597725806726839267657486659860, −11.66241987490309712350422301727, −11.00045261825189213920090753642, −9.96148863728943019766695745408, −9.46856899283261748198052887445, −8.65309652536055767685260941241, −7.59408800964439896553639333018, −6.573240890349548174218073068169, −5.84281877992938144018908142210, −4.48560067470653327617437252250, −3.10629150483684413144403689494, −2.62714839260327865930930388301, −1.6520882458500337128219971593,
0.15139972789979515952966750491, 1.48566154383468889033380408724, 2.47967484414022710323510579060, 4.12324894992422371434761458179, 5.18043807905235546738662542339, 5.77387920703361493351362697095, 7.10241009671005071557541780667, 7.36280234769790014670285859930, 8.66947705201126521072415615975, 9.45010041291863547797498391591, 10.08309889215877574539599531237, 10.70764802643149661385867172123, 12.12589464373083771811143096407, 13.15460350707810695480583845076, 13.669176216619520036867948627336, 14.69256967432230805065990184037, 15.640938713154273353636948579834, 16.220585367713322185544449634807, 17.26598126590886659803537500165, 17.5556454998250862662841786403, 18.30429868956988350460885112409, 19.63707931941944184443730858974, 20.04198960617036281330648863838, 20.6602413702111802775977820527, 22.04806952876220641159912340394