L(s) = 1 | + (0.0285 + 0.999i)2-s + (0.809 + 0.587i)3-s + (−0.998 + 0.0570i)4-s + (−0.516 − 0.856i)5-s + (−0.564 + 0.825i)6-s + (−0.0855 − 0.996i)8-s + (0.309 + 0.951i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (−0.254 + 0.967i)13-s + (0.0855 − 0.996i)15-s + (0.993 − 0.113i)16-s + (−0.985 − 0.170i)17-s + (−0.941 + 0.336i)18-s + (0.696 − 0.717i)19-s + (0.564 + 0.825i)20-s + ⋯ |
L(s) = 1 | + (0.0285 + 0.999i)2-s + (0.809 + 0.587i)3-s + (−0.998 + 0.0570i)4-s + (−0.516 − 0.856i)5-s + (−0.564 + 0.825i)6-s + (−0.0855 − 0.996i)8-s + (0.309 + 0.951i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (−0.254 + 0.967i)13-s + (0.0855 − 0.996i)15-s + (0.993 − 0.113i)16-s + (−0.985 − 0.170i)17-s + (−0.941 + 0.336i)18-s + (0.696 − 0.717i)19-s + (0.564 + 0.825i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1514616974 + 1.196444498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1514616974 + 1.196444498i\) |
\(L(1)\) |
\(\approx\) |
\(0.7868841823 + 0.7030052284i\) |
\(L(1)\) |
\(\approx\) |
\(0.7868841823 + 0.7030052284i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.0285 + 0.999i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.516 - 0.856i)T \) |
| 13 | \( 1 + (-0.254 + 0.967i)T \) |
| 17 | \( 1 + (-0.985 - 0.170i)T \) |
| 19 | \( 1 + (0.696 - 0.717i)T \) |
| 23 | \( 1 + (0.415 + 0.909i)T \) |
| 29 | \( 1 + (0.466 + 0.884i)T \) |
| 31 | \( 1 + (-0.774 - 0.633i)T \) |
| 37 | \( 1 + (-0.362 + 0.931i)T \) |
| 41 | \( 1 + (0.610 + 0.791i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 + (-0.941 - 0.336i)T \) |
| 53 | \( 1 + (0.993 + 0.113i)T \) |
| 59 | \( 1 + (-0.610 + 0.791i)T \) |
| 61 | \( 1 + (-0.0285 + 0.999i)T \) |
| 67 | \( 1 + (-0.959 - 0.281i)T \) |
| 71 | \( 1 + (0.897 + 0.441i)T \) |
| 73 | \( 1 + (-0.870 - 0.491i)T \) |
| 79 | \( 1 + (0.921 + 0.389i)T \) |
| 83 | \( 1 + (-0.736 - 0.676i)T \) |
| 89 | \( 1 + (0.142 + 0.989i)T \) |
| 97 | \( 1 + (-0.516 + 0.856i)T \) |
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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
−21.6477911544510240963781453679, −20.70886758164470725103250577103, −20.04331610578176143811932488202, −19.43863394809112516675193026877, −18.757152847938757794501001650, −18.03488633131538062274111096380, −17.48422772330704470741310606631, −15.8759124135932690573277441319, −14.977377574946153440240067678510, −14.334153356871371829369511858787, −13.621688859626499572521287834926, −12.638655906725612396933680992587, −12.150719379872805064463041724179, −11.0430690329707802424354459135, −10.394493528206033115433678500355, −9.43559469120355355944394424899, −8.48567042789495315365849185900, −7.785421770911383021514705316578, −6.88317860820069785513645373160, −5.70106182669546440132463935180, −4.3112659837616783809525164208, −3.47586017369757028741885051845, −2.73769068397196476997396401261, −1.94301543250936389576291888448, −0.50701482937090335305470118266,
1.39320473839313387030603693720, 2.97657854272533291040520918182, 4.08844233947404049458989488516, 4.657418717348610304385414620669, 5.417813230948111834821070539615, 6.86307486013656489491568595876, 7.56327127437638812869295290510, 8.46468950349271253105400217493, 9.20606333487042006761768416896, 9.54645872889329772389315566378, 10.98565766113356815246107884826, 12.01761910520541658803013425392, 13.27911785715747888765169832007, 13.53439150907905083040539610617, 14.680032374881491710570632470681, 15.27084521938364921655322349878, 16.12560455746060956299446077008, 16.4654621284382925906432251007, 17.46749215166602213423288921425, 18.43549565069694500971737213978, 19.53389580720105020093138187485, 19.855720914675309576173870620662, 21.01601455022052911325603046509, 21.696120467277064719373111591604, 22.46317124746701124215571899660