L(s) = 1 | + (0.350 − 0.936i)2-s + (0.851 + 0.523i)3-s + (−0.754 − 0.656i)4-s + (0.993 + 0.110i)5-s + (0.789 − 0.614i)6-s + (−0.0825 + 0.996i)7-s + (−0.879 + 0.475i)8-s + (0.451 + 0.892i)9-s + (0.451 − 0.892i)10-s + (−0.998 − 0.0550i)11-s + (−0.298 − 0.954i)12-s + (0.716 − 0.697i)13-s + (0.904 + 0.426i)14-s + (0.789 + 0.614i)15-s + (0.137 + 0.990i)16-s + (0.137 + 0.990i)17-s + ⋯ |
L(s) = 1 | + (0.350 − 0.936i)2-s + (0.851 + 0.523i)3-s + (−0.754 − 0.656i)4-s + (0.993 + 0.110i)5-s + (0.789 − 0.614i)6-s + (−0.0825 + 0.996i)7-s + (−0.879 + 0.475i)8-s + (0.451 + 0.892i)9-s + (0.451 − 0.892i)10-s + (−0.998 − 0.0550i)11-s + (−0.298 − 0.954i)12-s + (0.716 − 0.697i)13-s + (0.904 + 0.426i)14-s + (0.789 + 0.614i)15-s + (0.137 + 0.990i)16-s + (0.137 + 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.329575391 - 0.1331053796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.329575391 - 0.1331053796i\) |
\(L(1)\) |
\(\approx\) |
\(1.692447328 - 0.2681337408i\) |
\(L(1)\) |
\(\approx\) |
\(1.692447328 - 0.2681337408i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.350 - 0.936i)T \) |
| 3 | \( 1 + (0.851 + 0.523i)T \) |
| 5 | \( 1 + (0.993 + 0.110i)T \) |
| 7 | \( 1 + (-0.0825 + 0.996i)T \) |
| 11 | \( 1 + (-0.998 - 0.0550i)T \) |
| 13 | \( 1 + (0.716 - 0.697i)T \) |
| 17 | \( 1 + (0.137 + 0.990i)T \) |
| 19 | \( 1 + (0.851 + 0.523i)T \) |
| 23 | \( 1 + (-0.879 - 0.475i)T \) |
| 29 | \( 1 + (-0.962 - 0.272i)T \) |
| 31 | \( 1 + (0.789 - 0.614i)T \) |
| 37 | \( 1 + (0.716 + 0.697i)T \) |
| 41 | \( 1 + (0.993 + 0.110i)T \) |
| 43 | \( 1 + (0.451 - 0.892i)T \) |
| 47 | \( 1 + (-0.926 - 0.376i)T \) |
| 53 | \( 1 + (0.350 + 0.936i)T \) |
| 59 | \( 1 + (-0.0825 + 0.996i)T \) |
| 61 | \( 1 + (0.635 + 0.771i)T \) |
| 67 | \( 1 + (0.635 - 0.771i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.962 + 0.272i)T \) |
| 79 | \( 1 + (-0.191 - 0.981i)T \) |
| 83 | \( 1 + (-0.926 - 0.376i)T \) |
| 89 | \( 1 + (0.137 + 0.990i)T \) |
| 97 | \( 1 + (-0.754 - 0.656i)T \) |
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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
−23.49810698696551638154163956606, −22.69234202572165614258877168853, −21.469427324124709299340928846568, −20.88318747676795090464673852665, −20.14374917698649263007868260960, −18.83909934740167700426523595135, −17.99027938201323518638754775507, −17.620996567672275206957135094859, −16.24405742004298510151497495679, −15.88465465779886708263793191288, −14.36702808638611398020846330146, −14.06481252286225925649098323162, −13.26754314624700556101679602413, −12.85965868895137829513595800881, −11.3824246787213371166541305606, −9.84200317221767140519039049686, −9.37899011244683412486055911547, −8.24977833300345672330731689799, −7.41892411617678405927580880615, −6.74014901852554768800972648146, −5.72807002476977925680997064979, −4.64781390705630924323806625653, −3.53496751816961969287979430301, −2.521002099729599316138043653447, −1.065886937484256263682136167417,
1.59515406100041286516729856878, 2.49864930900325846124570771110, 3.13056628927879330686744375615, 4.29896584998286676734711089136, 5.58216831963529679264948394550, 5.88297400944854192388381456137, 7.97679642130121627392079828223, 8.68220878719312850571572814817, 9.67877190547107810835361237922, 10.19762990570484073564953222681, 11.0088583887956687737637652933, 12.329532438552262420798628319600, 13.185223748433228729295426954766, 13.67216973771784662042550705650, 14.74699523346946018371076625400, 15.283899972487503306029567254675, 16.333964259886848869440498939723, 17.79729307297044295091785616855, 18.4723191083018562137561721173, 19.041831106630699048334362520620, 20.2796571587749770725433203345, 20.797579764301209699663568542, 21.4369473581495138525139027062, 22.118632102012240940876727016722, 22.797955560285660308715963367753