L(s) = 1 | + (−0.593 − 0.805i)2-s + (−0.920 − 0.390i)3-s + (−0.296 + 0.955i)4-s + (−0.695 + 0.718i)5-s + (0.231 + 0.972i)6-s + (−0.0334 − 0.999i)7-s + (0.944 − 0.328i)8-s + (0.695 + 0.718i)9-s + (0.991 + 0.133i)10-s + (−0.296 − 0.955i)11-s + (0.645 − 0.763i)12-s + (0.231 + 0.972i)13-s + (−0.784 + 0.619i)14-s + (0.920 − 0.390i)15-s + (−0.824 − 0.565i)16-s + (−0.784 − 0.619i)17-s + ⋯ |
L(s) = 1 | + (−0.593 − 0.805i)2-s + (−0.920 − 0.390i)3-s + (−0.296 + 0.955i)4-s + (−0.695 + 0.718i)5-s + (0.231 + 0.972i)6-s + (−0.0334 − 0.999i)7-s + (0.944 − 0.328i)8-s + (0.695 + 0.718i)9-s + (0.991 + 0.133i)10-s + (−0.296 − 0.955i)11-s + (0.645 − 0.763i)12-s + (0.231 + 0.972i)13-s + (−0.784 + 0.619i)14-s + (0.920 − 0.390i)15-s + (−0.824 − 0.565i)16-s + (−0.784 − 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2189603924 + 0.06526674253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2189603924 + 0.06526674253i\) |
\(L(1)\) |
\(\approx\) |
\(0.3893900587 - 0.1889998116i\) |
\(L(1)\) |
\(\approx\) |
\(0.3893900587 - 0.1889998116i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (-0.593 - 0.805i)T \) |
| 3 | \( 1 + (-0.920 - 0.390i)T \) |
| 5 | \( 1 + (-0.695 + 0.718i)T \) |
| 7 | \( 1 + (-0.0334 - 0.999i)T \) |
| 11 | \( 1 + (-0.296 - 0.955i)T \) |
| 13 | \( 1 + (0.231 + 0.972i)T \) |
| 17 | \( 1 + (-0.784 - 0.619i)T \) |
| 19 | \( 1 + (-0.231 - 0.972i)T \) |
| 23 | \( 1 + (-0.741 + 0.670i)T \) |
| 29 | \( 1 + (-0.166 - 0.986i)T \) |
| 31 | \( 1 + (-0.480 - 0.876i)T \) |
| 37 | \( 1 + (-0.100 + 0.994i)T \) |
| 41 | \( 1 + (-0.944 - 0.328i)T \) |
| 43 | \( 1 + (0.979 + 0.199i)T \) |
| 47 | \( 1 + (-0.231 + 0.972i)T \) |
| 53 | \( 1 + (0.824 - 0.565i)T \) |
| 59 | \( 1 + (-0.997 + 0.0667i)T \) |
| 61 | \( 1 + (0.991 - 0.133i)T \) |
| 67 | \( 1 + (0.645 + 0.763i)T \) |
| 71 | \( 1 + (-0.296 - 0.955i)T \) |
| 73 | \( 1 + (0.480 - 0.876i)T \) |
| 79 | \( 1 + (0.979 - 0.199i)T \) |
| 83 | \( 1 + (0.359 + 0.933i)T \) |
| 89 | \( 1 + (-0.979 - 0.199i)T \) |
| 97 | \( 1 + (-0.741 + 0.670i)T \) |
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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
−25.142059216232301931450478086384, −24.434699890579526252519140752558, −23.46904598594310496530290231113, −22.867317597730680803036274402659, −21.91835267801844869568553735909, −20.58871523883249353837919397618, −19.762849264011805234116381714244, −18.43555680429239208663374246881, −17.92379587150536370377245941171, −16.91638859768186616615564404741, −16.013447138746409161502363508596, −15.460153900304690135720424377986, −14.73808041375679603279408285907, −12.78921930039494707154266316296, −12.291617558961890727085492881721, −10.93764018636866356275407346465, −10.09446379319469585856926476191, −8.95061608499577187691066406289, −8.157287057835869843232269208800, −6.94526700465288230964193023533, −5.75656515887241215430163923007, −5.10368265240326400876779457824, −4.01628526700549473131109659862, −1.70458310917434378794804008669, −0.15341873598968580007453650323,
0.72484908733137925235845147757, 2.26735783013002773480286135884, 3.7078691689215316953711479322, 4.58544040857127066820621985612, 6.43763318303153017676050560774, 7.253750580997286427098630848567, 8.1173347449521516329022937270, 9.584257488489985233550019976976, 10.753932136373422452673471618302, 11.25215597406056128352353006933, 11.84334657794374986314181857754, 13.333030178679834610712992701238, 13.778237801811293291544189074601, 15.70832931794450168546328072907, 16.47778024124490318691892057182, 17.37227776550688230831440979168, 18.25803609787001600902475823990, 19.058906245827872761598824590167, 19.661020014909941262515904828182, 20.86752973792134171048139697533, 21.982024610569102846416179306940, 22.51872982580268858264721135535, 23.63889591464298413559882866113, 24.144501201286426109946034716668, 25.899977782853536494492541424341