L(s) = 1 | + (0.559 + 0.829i)2-s + (0.990 + 0.139i)3-s + (−0.374 + 0.927i)4-s + (0.438 + 0.898i)6-s + (0.978 + 0.207i)7-s + (−0.978 + 0.207i)8-s + (0.961 + 0.275i)9-s + (−0.5 + 0.866i)12-s + (−0.719 + 0.694i)13-s + (0.374 + 0.927i)14-s + (−0.719 − 0.694i)16-s + (−0.961 + 0.275i)17-s + (0.309 + 0.951i)18-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.997 + 0.0697i)24-s + ⋯ |
L(s) = 1 | + (0.559 + 0.829i)2-s + (0.990 + 0.139i)3-s + (−0.374 + 0.927i)4-s + (0.438 + 0.898i)6-s + (0.978 + 0.207i)7-s + (−0.978 + 0.207i)8-s + (0.961 + 0.275i)9-s + (−0.5 + 0.866i)12-s + (−0.719 + 0.694i)13-s + (0.374 + 0.927i)14-s + (−0.719 − 0.694i)16-s + (−0.961 + 0.275i)17-s + (0.309 + 0.951i)18-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.997 + 0.0697i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.6802652683 + 2.602782925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.6802652683 + 2.602782925i\) |
\(L(1)\) |
\(\approx\) |
\(1.245536176 + 1.192697737i\) |
\(L(1)\) |
\(\approx\) |
\(1.245536176 + 1.192697737i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.559 + 0.829i)T \) |
| 3 | \( 1 + (0.990 + 0.139i)T \) |
| 7 | \( 1 + (0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.719 + 0.694i)T \) |
| 17 | \( 1 + (-0.961 + 0.275i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.615 + 0.788i)T \) |
| 31 | \( 1 + (-0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.990 - 0.139i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (-0.0348 - 0.999i)T \) |
| 53 | \( 1 + (-0.241 - 0.970i)T \) |
| 59 | \( 1 + (-0.0348 + 0.999i)T \) |
| 61 | \( 1 + (-0.997 - 0.0697i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.241 - 0.970i)T \) |
| 73 | \( 1 + (0.882 - 0.469i)T \) |
| 79 | \( 1 + (-0.438 + 0.898i)T \) |
| 83 | \( 1 + (0.104 + 0.994i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.559 + 0.829i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.765657546110610351416023602752, −20.0624843203593495241528688519, −19.78529857745661769778617716858, −18.70066702295095395127902151656, −18.031107075770527012134078245110, −17.30593606608648206484977584241, −15.71754614140197639834560959679, −15.16032051301745996198985134446, −14.38766133422422234538947792570, −13.79216763150571206528750457497, −13.0749995279632567199864550145, −12.229140530663013298218565572980, −11.3984931890485811453294648457, −10.5075909957875153858238377110, −9.73024766550355556097582574671, −8.90567175295305405876049204860, −7.994379956197362079596430635332, −7.19233344611736336110344887855, −5.92918950400104925002203395151, −4.83318994525334641799902271827, −4.20563177138497266509276369239, −3.20644621824777862262142935309, −2.26408266868623405766589902690, −1.62188414656815774077125028817, −0.33941324080134331761282827212,
1.72141212445831565919729004575, 2.5591146061335393566496824634, 3.67210277976065136420117177043, 4.580244586724956980268371810007, 5.05795045725319190729307316247, 6.48175129574559371297645529735, 7.16392944797157405190913838378, 8.13650031621631137821762719059, 8.60800938792117317038025926498, 9.42690600416747109288907921460, 10.56036061793805405608799678552, 11.73416324719063289319232341728, 12.46281041500243219875373380695, 13.466349108051191007048886295629, 14.051067706555486244467384021093, 14.83542914408438746138662505687, 15.16246665008365302286964724113, 16.21785603596675220711644091326, 16.8798472271307152768752958981, 18.01557801955905987432631021451, 18.39908802129203747660410867430, 19.68120020649621799576041222312, 20.31630618205045762104206189676, 21.25927220321732099895657404257, 21.74960966018187714666580108272