L(s) = 1 | + (0.996 − 0.0871i)5-s + (0.0871 − 0.996i)11-s + (−0.0871 − 0.996i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.996 − 0.0871i)29-s + (−0.766 − 0.642i)31-s + (−0.707 + 0.707i)37-s + (0.642 − 0.766i)41-s + (0.906 − 0.422i)43-s + (−0.766 + 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
L(s) = 1 | + (0.996 − 0.0871i)5-s + (0.0871 − 0.996i)11-s + (−0.0871 − 0.996i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.996 − 0.0871i)29-s + (−0.766 − 0.642i)31-s + (−0.707 + 0.707i)37-s + (0.642 − 0.766i)41-s + (0.906 − 0.422i)43-s + (−0.766 + 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07597124340 - 0.9601855115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07597124340 - 0.9601855115i\) |
\(L(1)\) |
\(\approx\) |
\(1.003964480 - 0.2908576744i\) |
\(L(1)\) |
\(\approx\) |
\(1.003964480 - 0.2908576744i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.996 - 0.0871i)T \) |
| 11 | \( 1 + (0.0871 - 0.996i)T \) |
| 13 | \( 1 + (-0.0871 - 0.996i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.965 + 0.258i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (-0.996 - 0.0871i)T \) |
| 31 | \( 1 + (-0.766 - 0.642i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (0.642 - 0.766i)T \) |
| 43 | \( 1 + (0.906 - 0.422i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (-0.258 - 0.965i)T \) |
| 59 | \( 1 + (0.573 + 0.819i)T \) |
| 61 | \( 1 + (0.996 + 0.0871i)T \) |
| 67 | \( 1 + (0.906 + 0.422i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.996 - 0.0871i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−17.854982616601485048389543102, −17.40413494840935383943203369205, −16.87705960723827005215452321276, −16.11745751354226457642352866886, −15.2844507874303441264896495205, −14.65207253585167166197330348725, −14.19107474909312477225175280117, −13.36942546951148085873829773306, −12.768106765917315184076682590518, −12.31705988363505946181287073096, −11.17277946573201887435582161188, −10.86143250123178395121408535939, −9.890434104845059167459319141257, −9.43488118696800723626327362649, −8.877238786143834738126759666552, −7.99719409698246318458252757694, −7.00822722324095902672481068227, −6.684405691618094844108533690689, −5.83407109442865096727988418874, −5.17882029835308045152307831172, −4.31286227148199498117802711090, −3.75587440804235332009315596160, −2.57398721878326036675511831992, −1.84759091064571851041740673357, −1.55586837619883718375822117156,
0.21894611846618579962777249466, 1.13718162261664752979592235767, 2.21856555871419787722643233004, 2.64524900665324912429502161764, 3.62202740535518021996120438095, 4.40272685843085293810111900949, 5.48134613095117835610781614303, 5.65521005469977042580605306637, 6.55067946280010247223230341321, 7.17724302319252534253426015306, 8.260563051655746332896332518222, 8.6274973174631261842157068131, 9.473123717394983853983076522500, 10.04828743182665546662251449555, 10.84026852951494242964915659873, 11.21943856691999753769181783397, 12.26852690198757338700507247765, 13.03010383279620534767168115395, 13.258759976394368265523676624920, 14.303807620987249725049249463896, 14.479159513163752165739698349825, 15.50771706590270478548386151748, 16.14538297895496455815097793618, 16.82488905153317745243115819286, 17.36837368164654747112249447865