L(s) = 1 | + (−0.850 − 0.526i)2-s + (0.932 + 0.361i)3-s + (0.445 + 0.895i)4-s + (−0.850 + 0.526i)5-s + (−0.602 − 0.798i)6-s + (−0.982 + 0.183i)7-s + (0.0922 − 0.995i)8-s + (0.739 + 0.673i)9-s + 10-s + (0.445 + 0.895i)11-s + (0.0922 + 0.995i)12-s + (−0.982 + 0.183i)13-s + (0.932 + 0.361i)14-s + (−0.982 + 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.0922 + 0.995i)17-s + ⋯ |
L(s) = 1 | + (−0.850 − 0.526i)2-s + (0.932 + 0.361i)3-s + (0.445 + 0.895i)4-s + (−0.850 + 0.526i)5-s + (−0.602 − 0.798i)6-s + (−0.982 + 0.183i)7-s + (0.0922 − 0.995i)8-s + (0.739 + 0.673i)9-s + 10-s + (0.445 + 0.895i)11-s + (0.0922 + 0.995i)12-s + (−0.982 + 0.183i)13-s + (0.932 + 0.361i)14-s + (−0.982 + 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.0922 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4819377846 + 0.4429468011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4819377846 + 0.4429468011i\) |
\(L(1)\) |
\(\approx\) |
\(0.7014835058 + 0.1796576414i\) |
\(L(1)\) |
\(\approx\) |
\(0.7014835058 + 0.1796576414i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (-0.850 - 0.526i)T \) |
| 3 | \( 1 + (0.932 + 0.361i)T \) |
| 5 | \( 1 + (-0.850 + 0.526i)T \) |
| 7 | \( 1 + (-0.982 + 0.183i)T \) |
| 11 | \( 1 + (0.445 + 0.895i)T \) |
| 13 | \( 1 + (-0.982 + 0.183i)T \) |
| 17 | \( 1 + (0.0922 + 0.995i)T \) |
| 19 | \( 1 + (-0.273 - 0.961i)T \) |
| 23 | \( 1 + (-0.602 + 0.798i)T \) |
| 29 | \( 1 + (-0.602 + 0.798i)T \) |
| 31 | \( 1 + (-0.273 + 0.961i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (-0.273 - 0.961i)T \) |
| 47 | \( 1 + (0.739 + 0.673i)T \) |
| 53 | \( 1 + (-0.273 - 0.961i)T \) |
| 59 | \( 1 + (0.739 + 0.673i)T \) |
| 61 | \( 1 + (0.739 - 0.673i)T \) |
| 67 | \( 1 + (-0.982 + 0.183i)T \) |
| 71 | \( 1 + (0.445 - 0.895i)T \) |
| 73 | \( 1 + (-0.982 - 0.183i)T \) |
| 79 | \( 1 + (0.932 - 0.361i)T \) |
| 83 | \( 1 + (0.0922 - 0.995i)T \) |
| 89 | \( 1 + (-0.850 + 0.526i)T \) |
| 97 | \( 1 + (0.445 + 0.895i)T \) |
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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
−28.00221163485444081324873507970, −26.85732774472093363870643855555, −26.55418484464593989169822836997, −25.14099331358703683675500884817, −24.6373620521488708137989441525, −23.694138660634789549742559114866, −22.52677421366561324726554509763, −20.68065557670679922374661201088, −19.886673674911879349971497322331, −19.1920181066680397108648110565, −18.49380991571818277428155868060, −16.79715423415620039424280419620, −16.184898752052859047859268435336, −15.084421342455098397957192329311, −14.1075896916162388185581555325, −12.78917753463144286866394690934, −11.617460727067708016754161103033, −9.93942495124779059067036835495, −9.14106877401413841128455375785, −8.05476227671273994777202117854, −7.2893752893769688134365286880, −6.02141959828550911421946063548, −4.09220328429964461973704222377, −2.61528364045656560178689990254, −0.67877200282695088954317582354,
2.14617473290641912700623200148, 3.268466861748026180867615628542, 4.22220787087649122698635594603, 6.86877926246088762271664594390, 7.601496716012889532833862272976, 8.91236123352019081022629587243, 9.74539734673350048299274164405, 10.69941737314031504892988004639, 12.10896552581777744306462875282, 12.966860975615084299451448925618, 14.675834803752481695868756755854, 15.50221698534004301542340676857, 16.450845951649497268253894806515, 17.79077760977818742851038979916, 19.16404401919506608520290644718, 19.55345231812093164848410593004, 20.21357684008385650951803517942, 21.764074534043300521795840992891, 22.25231009326558305913002006889, 23.89513932933989010960018553021, 25.36997151535717877729012553983, 25.90192002432854250924928604982, 26.74292524033977402468390469713, 27.611728031807433288202978227361, 28.46824161447259137625290646984