L(s) = 1 | + (−0.273 + 0.961i)2-s + (−0.982 + 0.183i)3-s + (−0.850 − 0.526i)4-s + (0.739 + 0.673i)5-s + (0.0922 − 0.995i)6-s + (−0.602 + 0.798i)7-s + (0.739 − 0.673i)8-s + (0.932 − 0.361i)9-s + (−0.850 + 0.526i)10-s + (−0.602 + 0.798i)11-s + (0.932 + 0.361i)12-s + (0.932 − 0.361i)13-s + (−0.602 − 0.798i)14-s + (−0.850 − 0.526i)15-s + (0.445 + 0.895i)16-s + (0.0922 − 0.995i)17-s + ⋯ |
L(s) = 1 | + (−0.273 + 0.961i)2-s + (−0.982 + 0.183i)3-s + (−0.850 − 0.526i)4-s + (0.739 + 0.673i)5-s + (0.0922 − 0.995i)6-s + (−0.602 + 0.798i)7-s + (0.739 − 0.673i)8-s + (0.932 − 0.361i)9-s + (−0.850 + 0.526i)10-s + (−0.602 + 0.798i)11-s + (0.932 + 0.361i)12-s + (0.932 − 0.361i)13-s + (−0.602 − 0.798i)14-s + (−0.850 − 0.526i)15-s + (0.445 + 0.895i)16-s + (0.0922 − 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7777300809 + 0.3500071988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7777300809 + 0.3500071988i\) |
\(L(1)\) |
\(\approx\) |
\(0.6308175522 + 0.3552650250i\) |
\(L(1)\) |
\(\approx\) |
\(0.6308175522 + 0.3552650250i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 953 | \( 1 \) |
good | 2 | \( 1 + (-0.273 + 0.961i)T \) |
| 3 | \( 1 + (-0.982 + 0.183i)T \) |
| 5 | \( 1 + (0.739 + 0.673i)T \) |
| 7 | \( 1 + (-0.602 + 0.798i)T \) |
| 11 | \( 1 + (-0.602 + 0.798i)T \) |
| 13 | \( 1 + (0.932 - 0.361i)T \) |
| 17 | \( 1 + (0.0922 - 0.995i)T \) |
| 19 | \( 1 + (0.932 - 0.361i)T \) |
| 23 | \( 1 + (0.445 - 0.895i)T \) |
| 29 | \( 1 + (0.0922 - 0.995i)T \) |
| 31 | \( 1 + (0.932 - 0.361i)T \) |
| 37 | \( 1 + (-0.273 - 0.961i)T \) |
| 41 | \( 1 + (-0.850 - 0.526i)T \) |
| 43 | \( 1 + (-0.273 - 0.961i)T \) |
| 47 | \( 1 + (0.0922 + 0.995i)T \) |
| 53 | \( 1 + (0.0922 - 0.995i)T \) |
| 59 | \( 1 + (0.0922 - 0.995i)T \) |
| 61 | \( 1 + (-0.602 - 0.798i)T \) |
| 67 | \( 1 + (0.932 - 0.361i)T \) |
| 71 | \( 1 + (0.932 - 0.361i)T \) |
| 73 | \( 1 + (-0.602 - 0.798i)T \) |
| 79 | \( 1 + (0.932 + 0.361i)T \) |
| 83 | \( 1 + (-0.602 - 0.798i)T \) |
| 89 | \( 1 + (0.739 - 0.673i)T \) |
| 97 | \( 1 + (-0.602 + 0.798i)T \) |
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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
−21.54375330082749518035455878921, −21.05443765299304908557243551324, −20.150743925473867575687704520435, −19.296977249604737256567492209067, −18.45562346051156159270177696063, −17.87913802787881681902122138378, −16.86342387494713487335674178638, −16.65158032967822892051684423600, −15.73079366918436820313463822499, −13.8787377915571636157229944058, −13.44359507614630188639727612917, −12.88720540234212498601358134027, −11.987651376266861698049267821602, −11.13781490624483191192534851429, −10.353192537158115296664337254947, −9.88555914811497543860988953275, −8.79626089935066900426895003742, −7.942730170941350708220020302634, −6.71937844303124528801270968377, −5.76847884129134487730335996583, −5.0118427326681631880912744806, −3.96148913323212689882520026609, −3.04761866755792724845898555549, −1.39669612065175344716387661516, −1.10415763480015147318704138469,
0.59384398248741658513053397703, 2.146061087784292074717968853964, 3.4323286165294207933917865222, 4.87898812980451654889487509976, 5.41976592732032030396250268334, 6.273503338135113290172570813898, 6.79479675087313180404247137256, 7.72559869903285147068719209061, 9.04594765943152823268846570411, 9.75239354315598006286472244474, 10.31369132706691665527126126546, 11.26939525837749005835761599066, 12.39782154401407484796681773774, 13.22738591786969846018914954501, 13.947096511459746318957244786137, 15.167481698739687656672868256675, 15.62897224322707677785853186950, 16.23028953179796740347052539510, 17.319379669125451228417511464418, 17.84012982551765399209554942553, 18.5633122889761117568359401579, 18.85849587748777102135647486976, 20.500336023512463542650084180511, 21.33377760152520149130301569988, 22.35724018406891677510743641692