L(s) = 1 | + (0.882 + 0.469i)2-s + (0.241 + 0.970i)3-s + (0.559 + 0.829i)4-s + (−0.241 + 0.970i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (−0.882 + 0.469i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.848 − 0.529i)13-s + (−0.0348 − 0.999i)14-s + (−0.374 + 0.927i)16-s + (0.438 + 0.898i)17-s − 18-s + (0.719 − 0.694i)21-s + (0.241 + 0.970i)22-s + ⋯ |
L(s) = 1 | + (0.882 + 0.469i)2-s + (0.241 + 0.970i)3-s + (0.559 + 0.829i)4-s + (−0.241 + 0.970i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (−0.882 + 0.469i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.848 − 0.529i)13-s + (−0.0348 − 0.999i)14-s + (−0.374 + 0.927i)16-s + (0.438 + 0.898i)17-s − 18-s + (0.719 − 0.694i)21-s + (0.241 + 0.970i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.6434248405 + 1.626656428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.6434248405 + 1.626656428i\) |
\(L(1)\) |
\(\approx\) |
\(1.073187219 + 1.021598475i\) |
\(L(1)\) |
\(\approx\) |
\(1.073187219 + 1.021598475i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.882 + 0.469i)T \) |
| 3 | \( 1 + (0.241 + 0.970i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.669 + 0.743i)T \) |
| 13 | \( 1 + (-0.848 - 0.529i)T \) |
| 17 | \( 1 + (0.438 + 0.898i)T \) |
| 23 | \( 1 + (-0.615 + 0.788i)T \) |
| 29 | \( 1 + (-0.438 + 0.898i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.374 - 0.927i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.438 - 0.898i)T \) |
| 53 | \( 1 + (-0.559 - 0.829i)T \) |
| 59 | \( 1 + (-0.990 + 0.139i)T \) |
| 61 | \( 1 + (-0.615 + 0.788i)T \) |
| 67 | \( 1 + (0.719 + 0.694i)T \) |
| 71 | \( 1 + (-0.961 - 0.275i)T \) |
| 73 | \( 1 + (0.848 - 0.529i)T \) |
| 79 | \( 1 + (0.241 + 0.970i)T \) |
| 83 | \( 1 + (0.913 + 0.406i)T \) |
| 89 | \( 1 + (0.374 + 0.927i)T \) |
| 97 | \( 1 + (0.719 - 0.694i)T \) |
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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
−23.01772718812408595552416378598, −22.22238648154428913281717771318, −21.60310810487367531582739194813, −20.41673737450978382521675618635, −19.714527645196348745380586079171, −18.82120169222021080258137703586, −18.53032153638553797315094232183, −16.993183821507115179824586808609, −16.07993512669354765862248158667, −14.91937930313748770648728121257, −14.21893318367240311373181671559, −13.51618771527581700724001034164, −12.44922234438628270625251406575, −11.99697071753441320597017414110, −11.22892806936501253921645434121, −9.73154141283279006192959339176, −8.98890723196922497640481820399, −7.65454771150113843600364475530, −6.54346810406898943147419982261, −5.99426228010994125132686885979, −4.853675088395949568904501350171, −3.41156363926995397934685166077, −2.63472777119316475757117163481, −1.66037698660148782561616206299, −0.284658118861487184050681019204,
2.02079500212646199587699619356, 3.46008235691925315966721039492, 3.90093596801487170998784365747, 4.96548859799099128177970144530, 5.87378032957299557167098187203, 7.10374633990292719411139720806, 7.84009472233428589804071970900, 9.141689574470721688631801449822, 10.09474811347683687549341041363, 10.91678006630013106336800233741, 12.10414988421387360929295790363, 12.92389703114437223821327495397, 13.990147514877055253798788737966, 14.69493102601074924924846794860, 15.31875091747583093186556925317, 16.36275999345702698538820997444, 16.94615858655205675935654491094, 17.66243036580614090419325581810, 19.64653774411482255819278396528, 19.96443449356211430727995587050, 20.853597109817745292261212476755, 21.862038956043460279213332573118, 22.36059350072924241957321866617, 23.14455812301540168249883281257, 23.96203254803801596800917825606