L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + 12-s + (−0.669 − 0.743i)13-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.809 + 0.587i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.978 − 0.207i)24-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + 12-s + (−0.669 − 0.743i)13-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.809 + 0.587i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.978 − 0.207i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.546676356 - 1.235333434i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.546676356 - 1.235333434i\) |
\(L(1)\) |
\(\approx\) |
\(2.383259808 - 0.4012193438i\) |
\(L(1)\) |
\(\approx\) |
\(2.383259808 - 0.4012193438i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.669 - 0.743i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.913 - 0.406i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.913 + 0.406i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.104 + 0.994i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.104 - 0.994i)T \) |
| 61 | \( 1 + (0.978 + 0.207i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.669 + 0.743i)T \) |
| 73 | \( 1 + (-0.104 + 0.994i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.978 + 0.207i)T \) |
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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
−21.48438924873460745985059939114, −21.14291809176717393464600959875, −20.01718618104272831782884724950, −19.38745421585505966575678367240, −18.907116236461932910935428353608, −17.68801500176715780146142809083, −16.66889624848034687642321065368, −15.94527765015631902590901415881, −14.99987365610635543181055600030, −14.68052402713281737929955138406, −13.623759706506791596971041994257, −13.1158429012356074766071533160, −12.23521486930201700081321184372, −11.79843749397828974917766027511, −10.381972289379856807002895932617, −9.49567224605937857862271837003, −8.61048173120913246674952767473, −7.65476638297425700200286604898, −6.89254342265620914671949095808, −6.16596674248768359178354725874, −5.16393012389077101850717752641, −4.00821774319322662210183972867, −3.26666548023163059451939637982, −2.46994172934029866581089639971, −1.564275435126215287206376026116,
1.06113479299507605218202883259, 2.56434807772268325698472889310, 3.01007220767159515452429603163, 3.92512273907973174595895574616, 4.756482516343312417858307283229, 5.637200892310960023607980853643, 6.92729472243813045175315741456, 7.38989223565822565003606800273, 8.53293380590285799048529691575, 9.73997993039679983414638970917, 10.19783842694503118958609310187, 10.98250120291490960400188663283, 12.32515013837940565920123939250, 12.78147474126096415743403102573, 13.717007305242530332976845902452, 14.26552659299514862536468231298, 14.98027993335210744217499506241, 15.938205656491993632973679753184, 16.28116056082619804471234206127, 17.39089277547348727675696767006, 18.820689501605304855899004932652, 19.39363200011595755116274547878, 20.12912541242504259295748619128, 20.62195180205938571660096542448, 21.43107912423466356294384552762