L(s) = 1 | + (0.690 + 0.723i)2-s + (0.866 − 0.5i)3-s + (−0.0475 + 0.998i)4-s + (0.959 + 0.281i)6-s + (−0.755 + 0.654i)8-s + (0.5 − 0.866i)9-s + (0.458 + 0.888i)12-s + (−0.540 + 0.841i)13-s + (−0.995 − 0.0950i)16-s + (0.618 + 0.786i)17-s + (0.971 − 0.235i)18-s + (−0.786 − 0.618i)19-s + (−0.0950 + 0.995i)23-s + (−0.327 + 0.945i)24-s + (−0.981 + 0.189i)26-s − i·27-s + ⋯ |
L(s) = 1 | + (0.690 + 0.723i)2-s + (0.866 − 0.5i)3-s + (−0.0475 + 0.998i)4-s + (0.959 + 0.281i)6-s + (−0.755 + 0.654i)8-s + (0.5 − 0.866i)9-s + (0.458 + 0.888i)12-s + (−0.540 + 0.841i)13-s + (−0.995 − 0.0950i)16-s + (0.618 + 0.786i)17-s + (0.971 − 0.235i)18-s + (−0.786 − 0.618i)19-s + (−0.0950 + 0.995i)23-s + (−0.327 + 0.945i)24-s + (−0.981 + 0.189i)26-s − i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.126787176 + 2.498991195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126787176 + 2.498991195i\) |
\(L(1)\) |
\(\approx\) |
\(1.579226618 + 0.8191173090i\) |
\(L(1)\) |
\(\approx\) |
\(1.579226618 + 0.8191173090i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.690 + 0.723i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.540 + 0.841i)T \) |
| 17 | \( 1 + (0.618 + 0.786i)T \) |
| 19 | \( 1 + (-0.786 - 0.618i)T \) |
| 23 | \( 1 + (-0.0950 + 0.995i)T \) |
| 29 | \( 1 + (0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.888 + 0.458i)T \) |
| 37 | \( 1 + (-0.998 + 0.0475i)T \) |
| 41 | \( 1 + (0.959 + 0.281i)T \) |
| 43 | \( 1 + (-0.755 + 0.654i)T \) |
| 47 | \( 1 + (-0.971 - 0.235i)T \) |
| 53 | \( 1 + (0.0950 + 0.995i)T \) |
| 59 | \( 1 + (0.723 + 0.690i)T \) |
| 61 | \( 1 + (-0.235 + 0.971i)T \) |
| 67 | \( 1 + (-0.971 + 0.235i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (0.814 + 0.580i)T \) |
| 79 | \( 1 + (0.327 + 0.945i)T \) |
| 83 | \( 1 + (0.909 + 0.415i)T \) |
| 89 | \( 1 + (0.928 + 0.371i)T \) |
| 97 | \( 1 + (-0.755 + 0.654i)T \) |
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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
−18.41668298720442312347934249253, −17.58181191814269267186828894429, −16.47756012422966400433356302225, −15.9845899285492878059043103584, −15.03083400107155808405268411421, −14.75227911996262151995759234387, −14.05171160472963539570008542602, −13.46211273519307778711715203839, −12.590147111383390745288496981278, −12.25678534221616382716786050683, −11.2064176879507453013384972618, −10.3991302317218146339972136115, −10.1140396569130105912304449297, −9.328446631936791217430386124700, −8.54807334719165694242278833481, −7.83287725251942492263354809043, −6.90449348583768262927828504927, −6.023177272160443906951531678488, −4.978175955947312149494046739412, −4.76101132909997537482549320572, −3.637009837600513393408210596153, −3.204123766252898685074531909551, −2.39183407396711208702244089229, −1.733106012979504105552160530717, −0.47921362861088165937395464408,
1.29143817600651510348898963872, 2.26506070785965697155120292187, 2.89414039407597067194561167554, 3.82687491248473194336460631072, 4.32952452585008995217905967344, 5.254874688252290104273148514350, 6.2386592745053922546772932648, 6.70872863647768169422974490998, 7.4806469601798226393352864560, 8.07376632477539616020930057392, 8.728125597322472821114254245049, 9.420505403466788024764480231201, 10.22488377023298281657924233060, 11.45880357545383112366442584099, 12.03848863784209812128188135688, 12.67261075985253682274503583716, 13.42413463442626345029226299867, 13.82287087565780576620229214223, 14.63360799203591059166590926204, 15.03838087181483131681948426646, 15.70001741294724763458325609279, 16.507843040410420637113816181240, 17.31353434529088172458048220223, 17.709547316261341012011830455107, 18.658922369740693617286074657813