L(s) = 1 | + (0.164 − 0.986i)2-s + (−0.0825 − 0.996i)3-s + (−0.945 − 0.324i)4-s + (−0.245 + 0.969i)5-s + (−0.996 − 0.0825i)6-s + (0.837 − 0.546i)7-s + (−0.475 + 0.879i)8-s + (−0.986 + 0.164i)9-s + (0.915 + 0.401i)10-s + (0.677 + 0.735i)11-s + (−0.245 + 0.969i)12-s + (0.969 + 0.245i)13-s + (−0.401 − 0.915i)14-s + (0.986 + 0.164i)15-s + (0.789 + 0.614i)16-s + (0.245 − 0.969i)17-s + ⋯ |
L(s) = 1 | + (0.164 − 0.986i)2-s + (−0.0825 − 0.996i)3-s + (−0.945 − 0.324i)4-s + (−0.245 + 0.969i)5-s + (−0.996 − 0.0825i)6-s + (0.837 − 0.546i)7-s + (−0.475 + 0.879i)8-s + (−0.986 + 0.164i)9-s + (0.915 + 0.401i)10-s + (0.677 + 0.735i)11-s + (−0.245 + 0.969i)12-s + (0.969 + 0.245i)13-s + (−0.401 − 0.915i)14-s + (0.986 + 0.164i)15-s + (0.789 + 0.614i)16-s + (0.245 − 0.969i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.293246897 - 1.483913952i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.293246897 - 1.483913952i\) |
\(L(1)\) |
\(\approx\) |
\(0.9493627062 - 0.7185450561i\) |
\(L(1)\) |
\(\approx\) |
\(0.9493627062 - 0.7185450561i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 229 | \( 1 \) |
good | 2 | \( 1 + (0.164 - 0.986i)T \) |
| 3 | \( 1 + (-0.0825 - 0.996i)T \) |
| 5 | \( 1 + (-0.245 + 0.969i)T \) |
| 7 | \( 1 + (0.837 - 0.546i)T \) |
| 11 | \( 1 + (0.677 + 0.735i)T \) |
| 13 | \( 1 + (0.969 + 0.245i)T \) |
| 17 | \( 1 + (0.245 - 0.969i)T \) |
| 19 | \( 1 + (0.245 + 0.969i)T \) |
| 23 | \( 1 + (0.915 - 0.401i)T \) |
| 29 | \( 1 + (-0.837 + 0.546i)T \) |
| 31 | \( 1 + (0.735 - 0.677i)T \) |
| 37 | \( 1 + (0.789 - 0.614i)T \) |
| 41 | \( 1 + (-0.164 + 0.986i)T \) |
| 43 | \( 1 + (0.789 - 0.614i)T \) |
| 47 | \( 1 + (0.164 + 0.986i)T \) |
| 53 | \( 1 + (-0.0825 - 0.996i)T \) |
| 59 | \( 1 + (-0.614 + 0.789i)T \) |
| 61 | \( 1 + (-0.986 + 0.164i)T \) |
| 67 | \( 1 + (0.164 + 0.986i)T \) |
| 71 | \( 1 + (0.677 - 0.735i)T \) |
| 73 | \( 1 + (0.475 - 0.879i)T \) |
| 79 | \( 1 + (-0.837 - 0.546i)T \) |
| 83 | \( 1 + (0.789 + 0.614i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.879 - 0.475i)T \) |
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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
−26.39707371463820181017226561886, −25.37860259124955559596269064836, −24.535168953324700022411139292033, −23.74248217885552447383835497946, −22.805915032489368940717774354762, −21.565714384793739674972219914649, −21.264279726834756589627985418916, −20.0058750815681846185387848323, −18.72722900144315514443768797567, −17.30131852697801433766659197168, −16.98190701276872684944019581335, −15.74476719562089457171252632992, −15.34027868766040892119151249102, −14.22856935393937939733501052148, −13.21926864612162921586895772059, −11.91504261919000663861957947094, −10.95446719080804534993937274756, −9.28833983267194597646854829801, −8.73838698865783095832414672271, −7.97613616414740231333297666078, −6.132226593953783785887261482595, −5.33598311322290533985006687633, −4.42892411468280962125973449420, −3.444623567817908598217440898681, −0.93468599325774281554379654695,
0.93960960819565236949373588348, 1.96076254052787042507365016267, 3.22266446032148225321512768721, 4.41414583748365410008283214538, 5.910358862737037791390471885436, 7.140230862757867072231516083885, 8.06106719383453031514088551743, 9.40080805050254274654678841949, 10.77642893143905088121030158759, 11.395063240488527582523526224076, 12.189222076971773925497702255391, 13.42368300536597172055192879561, 14.2324451208359921526289367023, 14.82507650343000857631428603443, 16.81212793151740603212053244333, 17.93419064078690744175831879255, 18.41074340062594177932016284716, 19.24816548710262768917388271256, 20.298236399779471042797488550431, 20.96290905196591756969828319887, 22.50276831244241677641387802243, 22.93635504381700963691281798504, 23.688543218011961856337467464428, 24.86852118956906860278043583845, 25.97138194776118882272567695984