L(s) = 1 | + (−0.0929 + 0.995i)2-s + (0.813 − 0.581i)3-s + (−0.982 − 0.185i)4-s + (0.346 − 0.938i)5-s + (0.503 + 0.863i)6-s + (0.995 + 0.0991i)7-s + (0.275 − 0.961i)8-s + (0.323 − 0.946i)9-s + (0.901 + 0.432i)10-s + (0.969 + 0.245i)11-s + (−0.907 + 0.421i)12-s + (0.299 + 0.954i)13-s + (−0.191 + 0.981i)14-s + (−0.263 − 0.964i)15-s + (0.931 + 0.363i)16-s + (−0.743 + 0.668i)17-s + ⋯ |
L(s) = 1 | + (−0.0929 + 0.995i)2-s + (0.813 − 0.581i)3-s + (−0.982 − 0.185i)4-s + (0.346 − 0.938i)5-s + (0.503 + 0.863i)6-s + (0.995 + 0.0991i)7-s + (0.275 − 0.961i)8-s + (0.323 − 0.946i)9-s + (0.901 + 0.432i)10-s + (0.969 + 0.245i)11-s + (−0.907 + 0.421i)12-s + (0.299 + 0.954i)13-s + (−0.191 + 0.981i)14-s + (−0.263 − 0.964i)15-s + (0.931 + 0.363i)16-s + (−0.743 + 0.668i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.940289652 + 0.1775261275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.940289652 + 0.1775261275i\) |
\(L(1)\) |
\(\approx\) |
\(1.438643944 + 0.2006795454i\) |
\(L(1)\) |
\(\approx\) |
\(1.438643944 + 0.2006795454i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.0929 + 0.995i)T \) |
| 3 | \( 1 + (0.813 - 0.581i)T \) |
| 5 | \( 1 + (0.346 - 0.938i)T \) |
| 7 | \( 1 + (0.995 + 0.0991i)T \) |
| 11 | \( 1 + (0.969 + 0.245i)T \) |
| 13 | \( 1 + (0.299 + 0.954i)T \) |
| 17 | \( 1 + (-0.743 + 0.668i)T \) |
| 19 | \( 1 + (-0.616 + 0.787i)T \) |
| 29 | \( 1 + (0.948 - 0.317i)T \) |
| 31 | \( 1 + (0.735 - 0.678i)T \) |
| 37 | \( 1 + (0.0558 - 0.998i)T \) |
| 41 | \( 1 + (-0.471 + 0.882i)T \) |
| 43 | \( 1 + (0.130 - 0.991i)T \) |
| 47 | \( 1 + (-0.775 + 0.631i)T \) |
| 53 | \( 1 + (0.901 - 0.432i)T \) |
| 59 | \( 1 + (-0.972 + 0.233i)T \) |
| 61 | \( 1 + (-0.977 - 0.209i)T \) |
| 67 | \( 1 + (0.179 + 0.983i)T \) |
| 71 | \( 1 + (-0.358 + 0.933i)T \) |
| 73 | \( 1 + (0.948 + 0.317i)T \) |
| 79 | \( 1 + (-0.426 - 0.904i)T \) |
| 83 | \( 1 + (-0.215 - 0.976i)T \) |
| 89 | \( 1 + (-0.635 - 0.771i)T \) |
| 97 | \( 1 + (0.0806 - 0.996i)T \) |
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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
−23.05207603802339299017748939164, −22.312631439375537852348404586976, −21.580692830442205498517114199256, −21.05198000543159255259972762221, −19.99049252040490727341295053391, −19.602884429711068971531162229464, −18.461676361650503846098422668008, −17.809649789511107762531565101431, −16.99122957187810503456139326103, −15.43377099511777505751581987358, −14.77858098728814702998157581614, −13.84787194564124761781648582428, −13.538354321328938830487685612404, −12.03719813374508762849275316652, −10.98585613319337418817866470825, −10.63422921431089235887448413297, −9.603453050298863429131449908039, −8.72364610429857057097613571230, −8.00094762770963232558876076925, −6.70479077259558485787954741914, −5.13171470528087204304125940619, −4.30578874773657202460202834419, −3.22059636592007023369958315229, −2.52000085677236531451390812005, −1.407805702221451295844422109742,
1.24257550583858745432676223679, 1.96580547246457736796188831349, 4.066978074604621289118527848462, 4.47961749758734549584267864874, 5.95073097375981873985349228333, 6.65062960542414518543751501694, 7.81903987360220929709578370896, 8.5984862077117329312307994689, 8.99087630656704446239728277452, 10.033149622415978900462440105470, 11.70893622660226322293235605621, 12.58258465698629257392018329837, 13.50588935714150414892277684721, 14.19838269722620511161627267423, 14.823696654356400353278453456770, 15.781421244957212948688774280478, 16.94990565801694443527210393352, 17.4197173992150112583945643788, 18.30026082034745506688197495964, 19.23396993596712131312964463519, 20.01109252497770777965504948435, 21.1157413571832632247580073931, 21.63014050365996480187135187915, 23.09662359798463475282159588439, 23.87286323866757643955430227405