L(s) = 1 | + (−0.235 + 0.971i)2-s + (−0.888 − 0.458i)4-s + (−0.928 − 0.371i)5-s + (0.654 − 0.755i)8-s + (0.580 − 0.814i)10-s + (0.235 + 0.971i)11-s + (−0.415 − 0.909i)13-s + (0.580 + 0.814i)16-s + (−0.0475 + 0.998i)17-s + (0.0475 + 0.998i)19-s + (0.654 + 0.755i)20-s − 22-s + (0.723 + 0.690i)25-s + (0.981 − 0.189i)26-s + (−0.841 − 0.540i)29-s + ⋯ |
L(s) = 1 | + (−0.235 + 0.971i)2-s + (−0.888 − 0.458i)4-s + (−0.928 − 0.371i)5-s + (0.654 − 0.755i)8-s + (0.580 − 0.814i)10-s + (0.235 + 0.971i)11-s + (−0.415 − 0.909i)13-s + (0.580 + 0.814i)16-s + (−0.0475 + 0.998i)17-s + (0.0475 + 0.998i)19-s + (0.654 + 0.755i)20-s − 22-s + (0.723 + 0.690i)25-s + (0.981 − 0.189i)26-s + (−0.841 − 0.540i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5266626674 - 0.1829710868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5266626674 - 0.1829710868i\) |
\(L(1)\) |
\(\approx\) |
\(0.6118284725 + 0.2380202189i\) |
\(L(1)\) |
\(\approx\) |
\(0.6118284725 + 0.2380202189i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.235 + 0.971i)T \) |
| 5 | \( 1 + (-0.928 - 0.371i)T \) |
| 11 | \( 1 + (0.235 + 0.971i)T \) |
| 13 | \( 1 + (-0.415 - 0.909i)T \) |
| 17 | \( 1 + (-0.0475 + 0.998i)T \) |
| 19 | \( 1 + (0.0475 + 0.998i)T \) |
| 29 | \( 1 + (-0.841 - 0.540i)T \) |
| 31 | \( 1 + (-0.981 - 0.189i)T \) |
| 37 | \( 1 + (0.786 + 0.618i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.995 + 0.0950i)T \) |
| 59 | \( 1 + (0.580 - 0.814i)T \) |
| 61 | \( 1 + (-0.327 - 0.945i)T \) |
| 67 | \( 1 + (-0.723 - 0.690i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.888 + 0.458i)T \) |
| 79 | \( 1 + (0.995 + 0.0950i)T \) |
| 83 | \( 1 + (0.142 - 0.989i)T \) |
| 89 | \( 1 + (-0.981 + 0.189i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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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
−23.6115710070119240578563383219, −22.50587075748131553798877014043, −21.96845601507967350438837234851, −21.105422861218439001278084504193, −19.99079553446585403732651017880, −19.525679761439457690100271215, −18.65848164372925298467266790483, −18.068080303015830057153343119, −16.740177372179079370524888599487, −16.170809442581239676169343739804, −14.82602108313471780718376724480, −14.02429671690845071811236952455, −13.101631345449350784001830874752, −11.99018074068477093057095226394, −11.35761157583025205514058557785, −10.8085738309761618324394778879, −9.43499184992007514836250310605, −8.83080586431498816525493539645, −7.69522490039201006933995976639, −6.84798122877792552301254819812, −5.22296897252082111929473164306, −4.1953073372760794485271411071, −3.31913678429134552828414438332, −2.3604034570851814697073197549, −0.829475363790983710137298944757,
0.22072148735592259573967398708, 1.62646919401081708945040714443, 3.58064405689497745623839800433, 4.42218923350930508883201076546, 5.40103006710681338553087653189, 6.45295361399803168119487412704, 7.72582541847157684774333373580, 7.907698819189818299653764201455, 9.170625460851831695100907780701, 10.014688939649163206254781332040, 11.09103210198206034856965862203, 12.53480388405013078027195984520, 12.820490854382284363675689891525, 14.33533822063910680516775822794, 15.05043943221700656687243921024, 15.599485544841568075897232195146, 16.67377196040481012865990452723, 17.255397112345959369529184825750, 18.23199965801203001396597892828, 19.15401359545819731668595365391, 19.9066188184032949230800881278, 20.74648447235291375616399404168, 22.24628945910124899796684562227, 22.75582137621349711059923323855, 23.6123275092642088927216211196