L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.587 + 0.809i)5-s + (0.587 − 0.809i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s − 12-s + (0.309 + 0.951i)14-s + (−0.951 − 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.951 + 0.309i)18-s + (−0.951 + 0.309i)19-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.587 + 0.809i)5-s + (0.587 − 0.809i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + 10-s − 12-s + (0.309 + 0.951i)14-s + (−0.951 − 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.951 + 0.309i)18-s + (−0.951 + 0.309i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09757521122 + 0.3012420649i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09757521122 + 0.3012420649i\) |
\(L(1)\) |
\(\approx\) |
\(0.5174869964 + 0.1348953962i\) |
\(L(1)\) |
\(\approx\) |
\(0.5174869964 + 0.1348953962i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.587 + 0.809i)T \) |
| 7 | \( 1 + (-0.951 - 0.309i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.951 + 0.309i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.587 + 0.809i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.951 + 0.309i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.951 + 0.309i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.587 + 0.809i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.587 + 0.809i)T \) |
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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
−28.11471775990339040571511642167, −26.64111620238856464196777508302, −25.81310494352691985485810990889, −24.950252477614441977930330389995, −24.10590509363255487644451843576, −23.440861402179706036892076128349, −22.35059866378109609894031391804, −20.4489736342052763376508946539, −19.4468706265814932767325285076, −19.07771676555217459386699116581, −17.76208052099799924404839788243, −16.85847631113988279349545905403, −15.78361517881970787556488544565, −14.91468225528083766958771339547, −13.44315256929190877778351998329, −12.76361039150661953776336698466, −11.44207849386228386782264912492, −9.718610188929876290999001373220, −8.68191368639199093047235326982, −7.96737532880500121315462716518, −6.7008668043453594551966537759, −5.8426527753393669541583610863, −4.128143023609645284722466253719, −2.09051473378380360374223555783, −0.30641406678351243620473119166,
2.53869593337235290418670994329, 3.49038793897420267840732244331, 4.42113434572875213957823366926, 6.589161663604450842663855750110, 7.931814786505785036254307345861, 9.08267253956069000790575907132, 10.14166381859181357383215172733, 10.7976794499695229026122279223, 11.88816012369276421552680230742, 13.26956697091734468893692644584, 14.45708710999114105680151861047, 15.78543211411105166509694133550, 16.45599404356637437031767776558, 17.781991820527847962342590251945, 19.00647432603917024791259768855, 19.739535213761694108463726113332, 20.49413955809013729892865533453, 21.83095465596309525447034461096, 22.35989673541820283914486443096, 23.31633135838748866587585829301, 25.394505948688641959620129057621, 26.0817683361475910419581203313, 26.84557144425213103107484348760, 27.53257008861977688088209415365, 28.59403255123445290748040571557