L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)12-s + (−0.809 − 0.587i)13-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (0.5 − 0.866i)18-s + (−0.669 + 0.743i)19-s + (−0.309 + 0.951i)22-s + (−0.913 − 0.406i)23-s + (0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)12-s + (−0.809 − 0.587i)13-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (0.5 − 0.866i)18-s + (−0.669 + 0.743i)19-s + (−0.309 + 0.951i)22-s + (−0.913 − 0.406i)23-s + (0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1004637897 - 0.2889982126i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1004637897 - 0.2889982126i\) |
\(L(1)\) |
\(\approx\) |
\(0.6622471822 + 0.01695747811i\) |
\(L(1)\) |
\(\approx\) |
\(0.6622471822 + 0.01695747811i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.669 + 0.743i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.978 - 0.207i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.913 - 0.406i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.104 - 0.994i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.978 + 0.207i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (-0.913 - 0.406i)T \) |
| 67 | \( 1 + (0.978 + 0.207i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.104 - 0.994i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−27.47399310646479817755733776093, −26.23427974482727058578846596562, −25.906872663269893839529836090976, −24.76438307436439795216663453246, −24.09984686268694244927792490579, −23.21993255826872368483925528053, −21.62199022689452922356231201065, −20.22304883320421487486049189345, −19.767388188915387391724977982183, −18.79099043393486874705604660976, −17.789472448513546467625956724583, −17.17683727565314910377844806329, −15.64630636386789893701557167507, −14.89701090207785360050614445476, −13.885809835606259636585067013513, −12.58166574192578198564582829259, −11.51612758709638102142335963949, −10.05631194321302338474396411018, −9.15066211817786702640955517163, −8.16023040787768327018322423274, −7.11286468257903345417755955107, −6.42009429722095356766996004506, −4.630620602639435410161905241734, −2.57863215701008563990178579982, −1.65360713482098550453636170061,
0.12483836550105255285748878788, 2.155490030012670426826392939585, 3.17534119905223603484695905208, 4.4469471101859149506011138569, 6.215260577627942811819672286190, 7.850081788450687246400434668829, 8.477658445469221099995326100488, 9.62667422398358045678132365846, 10.43521099830069625794269994535, 11.39208065460706547454681423482, 12.738109045299667514364048945951, 13.9635557281069424162558847043, 15.2174258421286470318487067567, 16.0971044309881115438592849772, 16.96968335724449388094234864599, 18.12201706962177162113125144434, 19.326180845827483525289027901862, 19.845123734802096807562054034771, 20.95123378619839136599968616155, 21.62838143974617662350616518211, 22.60558245850418974051333283152, 24.498959840434345451069980362979, 25.02295979585645316776323792949, 26.32021974461704915107192414151, 26.75268167588948104807180287926