L(s) = 1 | + (−0.988 − 0.149i)5-s + (0.733 + 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.900 + 0.433i)17-s − 19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.0747 − 0.997i)29-s + (0.5 + 0.866i)31-s + (−0.900 + 0.433i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (0.733 + 0.680i)47-s + (−0.900 − 0.433i)53-s + (−0.623 − 0.781i)55-s + ⋯ |
L(s) = 1 | + (−0.988 − 0.149i)5-s + (0.733 + 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.900 + 0.433i)17-s − 19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.0747 − 0.997i)29-s + (0.5 + 0.866i)31-s + (−0.900 + 0.433i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (0.733 + 0.680i)47-s + (−0.900 − 0.433i)53-s + (−0.623 − 0.781i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7315144942 - 0.7289135448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7315144942 - 0.7289135448i\) |
\(L(1)\) |
\(\approx\) |
\(0.8634850523 - 0.03843020987i\) |
\(L(1)\) |
\(\approx\) |
\(0.8634850523 - 0.03843020987i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.988 - 0.149i)T \) |
| 11 | \( 1 + (0.733 + 0.680i)T \) |
| 13 | \( 1 + (0.955 - 0.294i)T \) |
| 17 | \( 1 + (-0.900 + 0.433i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.0747 - 0.997i)T \) |
| 29 | \( 1 + (0.0747 - 0.997i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 41 | \( 1 + (-0.988 - 0.149i)T \) |
| 43 | \( 1 + (0.988 - 0.149i)T \) |
| 47 | \( 1 + (0.733 + 0.680i)T \) |
| 53 | \( 1 + (-0.900 - 0.433i)T \) |
| 59 | \( 1 + (-0.365 + 0.930i)T \) |
| 61 | \( 1 + (0.0747 - 0.997i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.222 - 0.974i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.955 - 0.294i)T \) |
| 89 | \( 1 + (-0.222 - 0.974i)T \) |
| 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
−20.12281883061416388117140168956, −19.458752077721234335794858302732, −18.88859825484892305153417691530, −18.17637198854065719787072463513, −17.191451265403734539627339114431, −16.52800501879819103509224065349, −15.646444845977423270725586306229, −15.325122131532423256557821211297, −14.21220023309768107716402894676, −13.67369499591502717386841926109, −12.711173578198889146601968138343, −11.90601712395928190006924839916, −11.110410316688528571298106221203, −10.85897450689948992146642117167, −9.50617766622828987894657666673, −8.70417423630491303303311992021, −8.23660329643508942389494029089, −7.10538173867625878450173610834, −6.56011989958804119846988434643, −5.62014864294228075351714966049, −4.44902672361684864491571508250, −3.840723670553868586628583144390, −3.097303147895408653855824244701, −1.85512896163180813981040810574, −0.77449606518234394124170978195,
0.253106408042297087339524617464, 1.34051135664320535599684800427, 2.39530366711468715954262817752, 3.57208040473597377369046105968, 4.22413724853025525269791642449, 4.8581339990644327865074784863, 6.26287235985054166179248603130, 6.689654532401684253039670541, 7.732325610957739765184195251056, 8.56334059693369344618662356187, 8.95189620046164449714913067038, 10.25504515541157979638933342650, 10.84786360670681000586344072149, 11.669758221082735624870453925052, 12.3955865078972978487701762914, 12.994030391885523664746301141239, 13.989357649903146296307711561758, 14.82165259375638338583717859980, 15.53021133476386510947972582748, 15.95330220484924396360974489295, 17.09059965009413025310472666235, 17.47312857791549258402535956471, 18.58585363800367428696452698813, 19.160524118141337485343286228180, 19.892886287253432897399696722709