L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5468665606 - 0.8970068537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5468665606 - 0.8970068537i\) |
\(L(1)\) |
\(\approx\) |
\(0.9175142161 - 0.3547929909i\) |
\(L(1)\) |
\(\approx\) |
\(0.9175142161 - 0.3547929909i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−23.75392114650284569629535125938, −23.01105773891876364794066525927, −22.262160813326000368839863779163, −21.38795117710788161077746672561, −20.693442751071452831046550869105, −19.58555755522934227289283682166, −18.73669236682980302288958392241, −18.01926162917867035720361979655, −17.23753673902227692982592870653, −16.251072550407766069094770020354, −15.15253449951928032273991300047, −14.549073353560806061625870885212, −13.63050236138058837145563622819, −12.75521026765990431190099988000, −11.62967039965465654831243736896, −10.85162574749981325529684583638, −9.78423677232356954636269824651, −9.27780769134159749920080828585, −7.7020818926042437149245371186, −7.07469651362378263257661945857, −6.078337126203615983383939825757, −5.00225385204544246847725257457, −3.88483577371129064084287308403, −2.579472560897660910585260801493, −1.83397141197649945471788530650,
0.53278654223515923904233391458, 1.96630860794996828811554477875, 3.08913416147578286506810671155, 4.45191973755098712081468981838, 5.34975136747483245156361283388, 6.155743196997879579470672961224, 7.4269958043488734202747123017, 8.56723175100130470207641067453, 9.0312867599261567226907251522, 10.37058950882203635405007939740, 10.94303138291120260321154224697, 12.35995867402027400405641135792, 12.96185366312020272536905864905, 13.6789854302409668413553455409, 14.85511188289402832320967412266, 15.672940881192499428565730790690, 16.71296739901352693464985936181, 17.25160841468394614354747375255, 18.20518010914182143303758676478, 19.18039908179619713236432440767, 20.13931605757503962473332362647, 20.730349035911772023504668852165, 21.82693618554948173480200367054, 22.187734057362916442000084896383, 23.81016081245432094185011866668