L(s) = 1 | + (−0.623 − 0.781i)3-s + (0.623 + 0.781i)5-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.222 − 0.974i)15-s + (−0.900 − 0.433i)17-s − 19-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s − 31-s + (0.623 − 0.781i)33-s + (−0.900 − 0.433i)37-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)3-s + (0.623 + 0.781i)5-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.222 − 0.974i)15-s + (−0.900 − 0.433i)17-s − 19-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s − 31-s + (0.623 − 0.781i)33-s + (−0.900 − 0.433i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08615545021 + 0.2902863003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08615545021 + 0.2902863003i\) |
\(L(1)\) |
\(\approx\) |
\(0.7446074823 + 0.008627649295i\) |
\(L(1)\) |
\(\approx\) |
\(0.7446074823 + 0.008627649295i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.222 + 0.974i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
| 17 | \( 1 + (-0.900 - 0.433i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 29 | \( 1 + (-0.900 - 0.433i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (0.623 + 0.781i)T \) |
| 43 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (-0.900 + 0.433i)T \) |
| 59 | \( 1 + (-0.623 + 0.781i)T \) |
| 61 | \( 1 + (-0.900 - 0.433i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.222 - 0.974i)T \) |
| 89 | \( 1 + (-0.222 + 0.974i)T \) |
| 97 | \( 1 + T \) |
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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
−26.47066695760310414772360423539, −25.52008494452145033378987109422, −24.21188774321766713853417613885, −23.720461802995032004207604892, −22.28427071235816587564803261148, −21.57601756498577023958205798638, −20.96350409311295913728208673550, −19.806667226589133281199898208516, −18.60564756747248296940601114781, −17.206903933325379429353300489454, −16.88421801393743571108294533590, −15.89264030773942205608929489742, −14.76017367863923306330514803786, −13.59115066163545389432103590254, −12.53569004555150240645443162111, −11.37093904562023735894910100920, −10.53844653927700894974911209324, −9.18065298545649684587111753943, −8.77200246845161722296501162323, −6.74649391077346618856647328391, −5.75303301148031185286697846444, −4.76365918107562221486101726099, −3.67566207999706292144552375675, −1.77141518808533125474735457593, −0.10658467733366176816935816744,
1.71936303951779836802085159291, 2.759955565962663017300699689201, 4.68711642051828558005975465262, 5.902889857689886301063714578301, 6.830611552971933587649117787552, 7.65417278193615440359257070244, 9.24218942529652255559552031382, 10.51597490925863027738783691461, 11.19936233350697582465210639872, 12.599005446282757681358818843411, 13.1933090194953815520930131217, 14.460205207082313881911941218731, 15.34049893940278784695034085778, 16.90139720460250478225029833817, 17.63163511023616875131252333887, 18.289674677271588721827326693747, 19.30340619225902683499548883987, 20.36989657464803266669899627401, 21.687374735881685739064196601357, 22.680534777124779037887167578176, 23.01544070524948069010857869390, 24.4483450713982583962892693904, 25.156736424400485687665949253091, 25.969420106951338367871157848828, 27.24179961689412496446524455070