L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 + 0.984i)4-s + (0.939 − 0.342i)6-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.866 − 0.5i)12-s + (−0.342 − 0.939i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s + i·18-s + (−0.766 + 0.642i)21-s + (−0.984 − 0.173i)23-s + (0.173 + 0.984i)24-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 + 0.984i)4-s + (0.939 − 0.342i)6-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.866 − 0.5i)12-s + (−0.342 − 0.939i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s + i·18-s + (−0.766 + 0.642i)21-s + (−0.984 − 0.173i)23-s + (0.173 + 0.984i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2153815286 + 0.4046516458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2153815286 + 0.4046516458i\) |
\(L(1)\) |
\(\approx\) |
\(0.6071070288 + 0.06862997235i\) |
\(L(1)\) |
\(\approx\) |
\(0.6071070288 + 0.06862997235i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.342 - 0.939i)T \) |
| 17 | \( 1 + (-0.642 - 0.766i)T \) |
| 23 | \( 1 + (-0.984 - 0.173i)T \) |
| 29 | \( 1 + (0.766 + 0.642i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.642 + 0.766i)T \) |
| 53 | \( 1 + (-0.984 - 0.173i)T \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.342 - 0.939i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.642 + 0.766i)T \) |
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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
−21.30330793726618783296603831862, −20.0308319851262583104497444169, −19.628630102207555853260990404799, −18.70063739972309335711740591282, −18.0629903109137309751529467314, −17.33518438829474529592998467733, −16.90666112746661727543680899400, −15.98999247546160001337376706250, −14.944099383873338900316029269958, −14.12692995881353231238410134529, −13.696482532928823897771728969161, −12.58927438752675642975481773799, −11.49948276681966606345056038247, −10.98837676600500585267524197839, −9.99520725996817780250258645821, −8.926691541940399445211543555004, −8.072438789663914810146263806429, −7.55068420000880813370014674976, −6.65094208331178643990872291935, −6.00903995754423411999577034210, −4.9699052196451038589316161862, −4.12502269238787129989649588614, −2.10957638408184681050201501162, −1.6371436961273354351826481310, −0.2615217631426757954886882467,
1.26088688327159970806662402862, 2.55816684431242367591200858430, 3.26494162249182913045194650672, 4.53072083962024095873507530588, 4.98563890570427677530298728849, 6.19581145484631885905558056667, 7.50268960760173217186692048475, 8.4044475619085083064413094757, 9.012279923107078241425316661660, 9.96341887709906440463161732814, 10.56574571090023759568397689134, 11.390937015857880962289994797149, 11.9459828348483004177857196165, 12.79622546853459360271306274682, 13.9978001690594129388827716354, 14.82062398964036350400590217740, 15.79125492665895635899174472755, 16.3261817533127348375544490755, 17.46545919278395703105375058836, 17.82036333587138135268150685158, 18.49163653283864948975178446280, 19.833217452020574194009296076961, 20.20774215468302300233475955992, 21.0189552283046803205033720482, 21.70984983439237720280062839301