L(s) = 1 | + (−0.374 + 0.927i)2-s + (0.961 + 0.275i)3-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)6-s + (−0.913 − 0.406i)7-s + (0.913 − 0.406i)8-s + (0.848 + 0.529i)9-s + (−0.5 − 0.866i)12-s + (0.0348 − 0.999i)13-s + (0.719 − 0.694i)14-s + (0.0348 + 0.999i)16-s + (−0.848 + 0.529i)17-s + (−0.809 + 0.587i)18-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.990 − 0.139i)24-s + ⋯ |
L(s) = 1 | + (−0.374 + 0.927i)2-s + (0.961 + 0.275i)3-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)6-s + (−0.913 − 0.406i)7-s + (0.913 − 0.406i)8-s + (0.848 + 0.529i)9-s + (−0.5 − 0.866i)12-s + (0.0348 − 0.999i)13-s + (0.719 − 0.694i)14-s + (0.0348 + 0.999i)16-s + (−0.848 + 0.529i)17-s + (−0.809 + 0.587i)18-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.990 − 0.139i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5284228957 + 1.259015773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5284228957 + 1.259015773i\) |
\(L(1)\) |
\(\approx\) |
\(0.8654966423 + 0.4371673335i\) |
\(L(1)\) |
\(\approx\) |
\(0.8654966423 + 0.4371673335i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.374 + 0.927i)T \) |
| 3 | \( 1 + (0.961 + 0.275i)T \) |
| 7 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (0.0348 - 0.999i)T \) |
| 17 | \( 1 + (-0.848 + 0.529i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (0.241 - 0.970i)T \) |
| 31 | \( 1 + (-0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.961 - 0.275i)T \) |
| 43 | \( 1 + (-0.173 + 0.984i)T \) |
| 47 | \( 1 + (0.997 - 0.0697i)T \) |
| 53 | \( 1 + (-0.882 + 0.469i)T \) |
| 59 | \( 1 + (0.997 + 0.0697i)T \) |
| 61 | \( 1 + (0.990 + 0.139i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (0.882 + 0.469i)T \) |
| 73 | \( 1 + (-0.559 + 0.829i)T \) |
| 79 | \( 1 + (0.615 + 0.788i)T \) |
| 83 | \( 1 + (0.978 - 0.207i)T \) |
| 89 | \( 1 + (0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.374 + 0.927i)T \) |
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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
−20.80785930278334314124791488944, −20.20652482129693121434678854410, −19.45155822687054091380867950766, −18.93712062049158270062846489811, −18.30306356463744093301390592717, −17.426409950751544933304159569044, −16.30884821762298192607595660959, −15.65981706080020513569788437501, −14.53192585667375566767063961222, −13.65626488650553460864267125563, −13.1871559475537941663686308228, −12.30242589091158090992577801227, −11.6281722697148889452852680482, −10.53993129851295491558023906993, −9.534503182408054481353628759539, −9.16591615193789411725822759922, −8.45354321539315913953842575444, −7.312302977599125470423094265355, −6.67343750688535105401663085629, −5.17605813562181395297720456162, −3.95295805071172408584357123116, −3.388775486774750140909101625748, −2.35460081478775980216636370288, −1.73907297870997090113393605770, −0.33918619310803071095649552029,
0.86484992134829413480303792628, 2.255321769619825435526945164233, 3.42676598599059887103071527579, 4.22158346117028477420144452600, 5.219100349033491633555609016309, 6.389193943565301478851898276947, 7.00985700840917873417240891975, 8.034608601488574961100972884850, 8.577103418697939980834168282834, 9.47291084492019036898998135392, 10.212290970854836995710461855080, 10.74266342015416257657097381402, 12.56559687536110711661158586519, 13.20221318179147703151266937034, 13.87104700725560328565450349730, 14.72203506731064554902742893689, 15.506142364406596546989052097249, 15.932078029375727660433589418438, 16.849443889408255373795100752681, 17.64053499790422100962645005884, 18.61057397904844317402284356181, 19.26910652737588660656055194285, 19.98540571923608207149847082843, 20.54132396753381166339214534188, 21.88974475882042884825498242302