L(s) = 1 | + (0.666 + 0.745i)2-s + (−0.176 − 0.984i)3-s + (−0.110 + 0.993i)4-s + (−0.525 − 0.850i)5-s + (0.616 − 0.787i)6-s + (0.0221 + 0.999i)7-s + (−0.814 + 0.580i)8-s + (−0.937 + 0.346i)9-s + (0.283 − 0.958i)10-s + (0.988 + 0.154i)11-s + (0.997 − 0.0663i)12-s + (−0.903 + 0.428i)13-s + (−0.730 + 0.683i)14-s + (−0.745 + 0.666i)15-s + (−0.975 − 0.219i)16-s + (−0.562 − 0.826i)17-s + ⋯ |
L(s) = 1 | + (0.666 + 0.745i)2-s + (−0.176 − 0.984i)3-s + (−0.110 + 0.993i)4-s + (−0.525 − 0.850i)5-s + (0.616 − 0.787i)6-s + (0.0221 + 0.999i)7-s + (−0.814 + 0.580i)8-s + (−0.937 + 0.346i)9-s + (0.283 − 0.958i)10-s + (0.988 + 0.154i)11-s + (0.997 − 0.0663i)12-s + (−0.903 + 0.428i)13-s + (−0.730 + 0.683i)14-s + (−0.745 + 0.666i)15-s + (−0.975 − 0.219i)16-s + (−0.562 − 0.826i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09106751797 + 0.5363643814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09106751797 + 0.5363643814i\) |
\(L(1)\) |
\(\approx\) |
\(0.8746320945 + 0.2668173883i\) |
\(L(1)\) |
\(\approx\) |
\(0.8746320945 + 0.2668173883i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 569 | \( 1 \) |
good | 2 | \( 1 + (0.666 + 0.745i)T \) |
| 3 | \( 1 + (-0.176 - 0.984i)T \) |
| 5 | \( 1 + (-0.525 - 0.850i)T \) |
| 7 | \( 1 + (0.0221 + 0.999i)T \) |
| 11 | \( 1 + (0.988 + 0.154i)T \) |
| 13 | \( 1 + (-0.903 + 0.428i)T \) |
| 17 | \( 1 + (-0.562 - 0.826i)T \) |
| 19 | \( 1 + (-0.993 + 0.110i)T \) |
| 23 | \( 1 + (-0.980 - 0.197i)T \) |
| 29 | \( 1 + (0.826 + 0.562i)T \) |
| 31 | \( 1 + (-0.945 + 0.325i)T \) |
| 37 | \( 1 + (0.999 - 0.0221i)T \) |
| 41 | \( 1 + (-0.197 + 0.980i)T \) |
| 43 | \( 1 + (-0.666 + 0.745i)T \) |
| 47 | \( 1 + (-0.467 - 0.883i)T \) |
| 53 | \( 1 + (-0.616 + 0.787i)T \) |
| 59 | \( 1 + (0.467 + 0.883i)T \) |
| 61 | \( 1 + (-0.996 + 0.0883i)T \) |
| 67 | \( 1 + (-0.562 + 0.826i)T \) |
| 71 | \( 1 + (-0.814 - 0.580i)T \) |
| 73 | \( 1 + (0.993 - 0.110i)T \) |
| 79 | \( 1 + (-0.240 + 0.970i)T \) |
| 83 | \( 1 + (-0.0442 - 0.999i)T \) |
| 89 | \( 1 + (0.945 + 0.325i)T \) |
| 97 | \( 1 + (-0.894 - 0.448i)T \) |
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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
−22.57930564082870450955818265773, −22.10420461310278001471663398563, −21.487504666512495418144061902888, −20.293285205222660925665616202150, −19.76176889655212858736604747740, −19.21853727482203799549626638461, −17.76876049719648576716633773932, −17.05576807714775419675428113940, −15.91363063225439794397254306065, −14.95175770247676679064427820687, −14.568053488823475467751535750473, −13.71693359793810701362704460651, −12.45915345144525133585079077746, −11.55651276695750267086571792233, −10.83917425308605071298622388330, −10.271641200171493373559883274601, −9.469884604531447926201820064162, −8.14485332509802128603562070144, −6.74362777579000188294777051464, −6.02673927078415767362497434367, −4.596866581995647545729673853856, −4.01822052489723269294219674682, −3.343716766837369288041399850922, −2.100511380140462249883496975100, −0.21365660640570787960955563230,
1.76522960336761941063554341409, 2.84431528693533746281221500784, 4.319843803462635321830899164917, 5.05177321979782665109004561574, 6.120404771786832736454530346388, 6.83935497886839378679998289095, 7.83417676395330581806313566728, 8.64757299692938709198880826324, 9.29658853723347204207249294630, 11.45399878556110136261059781363, 11.99935405556966892600189179150, 12.51270471683564591095488079350, 13.34298627766177711836635859388, 14.43220618610250800007406061082, 15.00174628579564841041508763141, 16.244915563722598107127879420649, 16.71427098700500031688694724934, 17.69271340557886729366391740864, 18.41231055218178626496874637688, 19.6254090091662565626552447387, 20.09960679289205851958829380784, 21.49925873551591261047901380826, 22.09748167603784548095949447366, 23.02133728081227239779137601135, 23.78772859547160692788557133819