L(s) = 1 | + (0.203 − 0.979i)2-s + (0.682 − 0.730i)3-s + (−0.917 − 0.398i)4-s + (−0.962 − 0.269i)5-s + (−0.576 − 0.816i)6-s + (−0.775 + 0.631i)7-s + (−0.576 + 0.816i)8-s + (−0.0682 − 0.997i)9-s + (−0.460 + 0.887i)10-s + (0.334 − 0.942i)11-s + (−0.917 + 0.398i)12-s + (0.990 − 0.136i)13-s + (0.460 + 0.887i)14-s + (−0.854 + 0.519i)15-s + (0.682 + 0.730i)16-s + (−0.334 − 0.942i)17-s + ⋯ |
L(s) = 1 | + (0.203 − 0.979i)2-s + (0.682 − 0.730i)3-s + (−0.917 − 0.398i)4-s + (−0.962 − 0.269i)5-s + (−0.576 − 0.816i)6-s + (−0.775 + 0.631i)7-s + (−0.576 + 0.816i)8-s + (−0.0682 − 0.997i)9-s + (−0.460 + 0.887i)10-s + (0.334 − 0.942i)11-s + (−0.917 + 0.398i)12-s + (0.990 − 0.136i)13-s + (0.460 + 0.887i)14-s + (−0.854 + 0.519i)15-s + (0.682 + 0.730i)16-s + (−0.334 − 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1218918698 - 1.177345140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1218918698 - 1.177345140i\) |
\(L(1)\) |
\(\approx\) |
\(0.5929139897 - 0.8216356878i\) |
\(L(1)\) |
\(\approx\) |
\(0.5929139897 - 0.8216356878i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (0.203 - 0.979i)T \) |
| 3 | \( 1 + (0.682 - 0.730i)T \) |
| 5 | \( 1 + (-0.962 - 0.269i)T \) |
| 7 | \( 1 + (-0.775 + 0.631i)T \) |
| 11 | \( 1 + (0.334 - 0.942i)T \) |
| 13 | \( 1 + (0.990 - 0.136i)T \) |
| 17 | \( 1 + (-0.334 - 0.942i)T \) |
| 19 | \( 1 + (-0.962 + 0.269i)T \) |
| 23 | \( 1 + (-0.203 - 0.979i)T \) |
| 29 | \( 1 + (0.990 + 0.136i)T \) |
| 31 | \( 1 + (-0.682 - 0.730i)T \) |
| 37 | \( 1 + (0.460 - 0.887i)T \) |
| 41 | \( 1 + (0.576 + 0.816i)T \) |
| 43 | \( 1 + (0.917 + 0.398i)T \) |
| 53 | \( 1 + (-0.576 - 0.816i)T \) |
| 59 | \( 1 + (-0.917 + 0.398i)T \) |
| 61 | \( 1 + (0.460 + 0.887i)T \) |
| 67 | \( 1 + (0.775 + 0.631i)T \) |
| 71 | \( 1 + (0.203 + 0.979i)T \) |
| 73 | \( 1 + (0.0682 - 0.997i)T \) |
| 79 | \( 1 + (0.854 - 0.519i)T \) |
| 83 | \( 1 + (-0.334 + 0.942i)T \) |
| 89 | \( 1 + (0.962 + 0.269i)T \) |
| 97 | \( 1 + (0.682 - 0.730i)T \) |
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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
−34.23683780182612125873142331611, −33.01547074077708643539069559159, −32.40125718505968667947917447355, −31.12663253351899657683323332160, −30.45410604492584402286258766550, −28.06419942774514859997322080934, −27.178971299594985910373314973173, −26.00848764098621803456180249039, −25.542413486337256202512907336064, −23.68943425880625015540901814775, −22.89750521464894718730332866427, −21.69373230215688498707930341918, −20.065412369473588484110420002840, −19.06735484711205059562441487935, −17.22833217260014598473119726338, −15.92150281776180142334785438955, −15.29043823767866198708296280750, −14.05282560036798041296920439715, −12.729916263610924775649633309089, −10.60707381930576907695617603196, −9.12997071731145234213078216048, −7.88540628164554380906819846103, −6.58519134342189345706720739670, −4.37224819235509026690858213360, −3.57066013974566632707521118042,
0.63175855690870534422657142295, 2.730409298073386134600771707111, 3.93350257849576094131322227614, 6.21392939354554593651953220302, 8.31544950175962087695848685029, 9.138176823569226918962954940054, 11.16762394147771653484884063701, 12.3415492911283555107103939971, 13.236417048587677347913225940119, 14.59647715831398320443487677822, 16.09184911646469929073034793904, 18.295062238402518624282078636819, 19.083773924939459446499773438240, 19.904477074876565843057365577292, 21.06018370072360104685829009330, 22.61765256987173577456703877512, 23.62715880114517675135870362493, 24.85928306852880625717650494464, 26.36372139084297399958456996824, 27.55802888019914414296169606514, 28.77683445125979005625949284990, 29.835290124272204021693204195627, 30.92407042799872480787925724682, 31.7552984156854697395576287821, 32.453500046698524723431819988117