L(s) = 1 | + (0.438 − 0.898i)2-s + (−0.374 − 0.927i)3-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)6-s + (0.978 + 0.207i)7-s + (−0.978 + 0.207i)8-s + (−0.719 + 0.694i)9-s + (−0.5 + 0.866i)12-s + (−0.241 − 0.970i)13-s + (0.615 − 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.719 + 0.694i)17-s + (0.309 + 0.951i)18-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.559 + 0.829i)24-s + ⋯ |
L(s) = 1 | + (0.438 − 0.898i)2-s + (−0.374 − 0.927i)3-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)6-s + (0.978 + 0.207i)7-s + (−0.978 + 0.207i)8-s + (−0.719 + 0.694i)9-s + (−0.5 + 0.866i)12-s + (−0.241 − 0.970i)13-s + (0.615 − 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.719 + 0.694i)17-s + (0.309 + 0.951i)18-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.559 + 0.829i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.192148416 - 2.037698886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.192148416 - 2.037698886i\) |
\(L(1)\) |
\(\approx\) |
\(0.8957804923 - 0.8519249112i\) |
\(L(1)\) |
\(\approx\) |
\(0.8957804923 - 0.8519249112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.438 - 0.898i)T \) |
| 3 | \( 1 + (-0.374 - 0.927i)T \) |
| 7 | \( 1 + (0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.241 - 0.970i)T \) |
| 17 | \( 1 + (0.719 + 0.694i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.990 + 0.139i)T \) |
| 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 | \( 1 + (0.882 + 0.469i)T \) |
| 53 | \( 1 + (0.961 + 0.275i)T \) |
| 59 | \( 1 + (0.882 - 0.469i)T \) |
| 61 | \( 1 + (0.559 - 0.829i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.961 + 0.275i)T \) |
| 73 | \( 1 + (-0.848 - 0.529i)T \) |
| 79 | \( 1 + (0.997 - 0.0697i)T \) |
| 83 | \( 1 + (0.104 + 0.994i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.438 - 0.898i)T \) |
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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
−21.710849691108964063468809938012, −20.929470002517289183410234229724, −20.6084183733102087689074060327, −19.12495306172717913267081414335, −18.16330248766440807380481516794, −17.41901207930019152445742033747, −16.73397147411444102467574943611, −16.23250680681608817558925734785, −15.26084199898374969249290339022, −14.57441279035327479424616902086, −14.141820818800973842625558217735, −13.07139852129379025754091663653, −11.900647620498291643367006458340, −11.44698036847266839396594642798, −10.41695265022946217897371033636, −9.226010448509043000288763520145, −8.89205101616619048949133066754, −7.56570606865060574711264954141, −7.05232990764350276046045916757, −5.63546829118761961446786353359, −5.32690976618240448561281258378, −4.2617299492253339697003317577, −3.78109897456923723092429206987, −2.421608109962973262347025937288, −0.68879599516942418872330185861,
0.71896729535413616456908214558, 1.47537323387803871375417947840, 2.39965684927812736029947789499, 3.327089611522260031736188119710, 4.63243869198005791578767781680, 5.42428451539132397933881162138, 6.00862847708473208811323917590, 7.32329857477850810164508145371, 8.132519746500253144342140481942, 8.96497684762240997950652634096, 10.20270665269629618864492988961, 11.000515693390244979172642032484, 11.49586649396575722128277285967, 12.60765164373113188151462349814, 12.758580158722185488448144651932, 13.84750227760270124484284188091, 14.639154806145189532952544195170, 15.16113032641235414166625371185, 16.69587383403209119081003273778, 17.482841716331162509187460130306, 18.151132621084862766774755808778, 18.77421212366569722569712014909, 19.52820928976206032629194735561, 20.35971812351163164974274444539, 20.99894036206844089101113591213