L(s) = 1 | + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (0.0475 + 0.998i)5-s + (0.959 − 0.281i)8-s + (0.786 − 0.618i)10-s + (0.580 − 0.814i)11-s + (−0.142 − 0.989i)13-s + (−0.786 − 0.618i)16-s + (0.981 + 0.189i)17-s + (−0.981 + 0.189i)19-s + (−0.959 − 0.281i)20-s − 22-s + (−0.995 + 0.0950i)25-s + (−0.723 + 0.690i)26-s + (0.654 − 0.755i)29-s + ⋯ |
L(s) = 1 | + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (0.0475 + 0.998i)5-s + (0.959 − 0.281i)8-s + (0.786 − 0.618i)10-s + (0.580 − 0.814i)11-s + (−0.142 − 0.989i)13-s + (−0.786 − 0.618i)16-s + (0.981 + 0.189i)17-s + (−0.981 + 0.189i)19-s + (−0.959 − 0.281i)20-s − 22-s + (−0.995 + 0.0950i)25-s + (−0.723 + 0.690i)26-s + (0.654 − 0.755i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9538172648 - 0.3037628459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9538172648 - 0.3037628459i\) |
\(L(1)\) |
\(\approx\) |
\(0.8217430288 - 0.1944051322i\) |
\(L(1)\) |
\(\approx\) |
\(0.8217430288 - 0.1944051322i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.580 - 0.814i)T \) |
| 5 | \( 1 + (0.0475 + 0.998i)T \) |
| 11 | \( 1 + (0.580 - 0.814i)T \) |
| 13 | \( 1 + (-0.142 - 0.989i)T \) |
| 17 | \( 1 + (0.981 + 0.189i)T \) |
| 19 | \( 1 + (-0.981 + 0.189i)T \) |
| 29 | \( 1 + (0.654 - 0.755i)T \) |
| 31 | \( 1 + (0.723 + 0.690i)T \) |
| 37 | \( 1 + (0.888 - 0.458i)T \) |
| 41 | \( 1 + (-0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.959 + 0.281i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.928 - 0.371i)T \) |
| 59 | \( 1 + (0.786 - 0.618i)T \) |
| 61 | \( 1 + (-0.235 + 0.971i)T \) |
| 67 | \( 1 + (0.995 - 0.0950i)T \) |
| 71 | \( 1 + (-0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.327 + 0.945i)T \) |
| 79 | \( 1 + (-0.928 - 0.371i)T \) |
| 83 | \( 1 + (0.841 + 0.540i)T \) |
| 89 | \( 1 + (0.723 - 0.690i)T \) |
| 97 | \( 1 + (-0.841 + 0.540i)T \) |
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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
−23.83239973688083033205754141095, −23.45093123387656738399582676201, −22.323197149782355990444149104748, −21.20275342365597834875755954928, −20.30956338113354405605890803881, −19.45643954699141528246846160790, −18.74022478454781001996952587368, −17.56711228704765050423934180329, −16.957626388039653413876287441479, −16.355085139272586845057931480364, −15.3704826240790932525262136333, −14.49426348571074511307043136330, −13.667693089024235766660226048279, −12.54326478800450874957603864358, −11.691674896644352102457061240295, −10.33050159419030706662524253811, −9.45897843188049012823739408391, −8.83553948500553874276181856271, −7.86372897704021161488120660425, −6.88904699727641999098607450063, −5.9554987748047131723582628744, −4.81939790462766926984048015737, −4.1778314064753532328607793041, −2.07114498223203549663933842819, −1.01174201557543611728828673771,
0.919629897879572372044045090585, 2.390651997419119369387490759723, 3.22029844205420562256625636484, 4.101006663769077931950607317656, 5.706698207937698405744716696630, 6.77361053953905213136303235688, 7.885530428359926447384592296744, 8.58523843923603132744149907963, 9.91340367173995575781512362563, 10.41431537821988031484987344140, 11.31761863362484886712696734178, 12.118519430192020395842405673540, 13.14829036896692938728591042033, 14.094808576521271010478883446993, 14.96029808519916589447253852648, 16.14565157802520734182055625749, 17.145462305179961006240746225013, 17.801252518289522136009610339111, 18.81405676408610202659653756828, 19.25674921706689507030059357788, 20.124421846655562597207291920183, 21.33386618089227456201117509194, 21.70014822124312930230046826639, 22.742527774248997731164572402751, 23.30160581836964481325755355334