L(s) = 1 | + (0.899 − 0.435i)2-s + (0.809 − 0.587i)3-s + (0.619 − 0.784i)4-s + (−0.594 − 0.804i)5-s + (0.471 − 0.881i)6-s + (0.339 − 0.940i)7-s + (0.215 − 0.976i)8-s + (0.309 − 0.951i)9-s + (−0.885 − 0.464i)10-s + (0.0402 − 0.999i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.953 − 0.301i)15-s + (−0.231 − 0.972i)16-s + (−0.932 − 0.362i)17-s + (−0.136 − 0.990i)18-s + ⋯ |
L(s) = 1 | + (0.899 − 0.435i)2-s + (0.809 − 0.587i)3-s + (0.619 − 0.784i)4-s + (−0.594 − 0.804i)5-s + (0.471 − 0.881i)6-s + (0.339 − 0.940i)7-s + (0.215 − 0.976i)8-s + (0.309 − 0.951i)9-s + (−0.885 − 0.464i)10-s + (0.0402 − 0.999i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.953 − 0.301i)15-s + (−0.231 − 0.972i)16-s + (−0.932 − 0.362i)17-s + (−0.136 − 0.990i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.9955625350 - 3.669346411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.9955625350 - 3.669346411i\) |
\(L(1)\) |
\(\approx\) |
\(1.290074782 - 1.712317124i\) |
\(L(1)\) |
\(\approx\) |
\(1.290074782 - 1.712317124i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (0.899 - 0.435i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.594 - 0.804i)T \) |
| 7 | \( 1 + (0.339 - 0.940i)T \) |
| 13 | \( 1 + (0.978 - 0.207i)T \) |
| 17 | \( 1 + (-0.932 - 0.362i)T \) |
| 19 | \( 1 + (-0.759 + 0.650i)T \) |
| 23 | \( 1 + (0.919 + 0.391i)T \) |
| 29 | \( 1 + (0.681 - 0.732i)T \) |
| 31 | \( 1 + (0.607 + 0.794i)T \) |
| 37 | \( 1 + (-0.974 - 0.223i)T \) |
| 41 | \( 1 + (0.913 + 0.406i)T \) |
| 43 | \( 1 + (-0.845 - 0.534i)T \) |
| 47 | \( 1 + (-0.999 - 0.0161i)T \) |
| 53 | \( 1 + (0.870 + 0.493i)T \) |
| 59 | \( 1 + (-0.759 - 0.650i)T \) |
| 61 | \( 1 + (-0.471 + 0.881i)T \) |
| 67 | \( 1 + (-0.845 - 0.534i)T \) |
| 71 | \( 1 + (-0.991 - 0.128i)T \) |
| 73 | \( 1 + (0.581 + 0.813i)T \) |
| 79 | \( 1 + (-0.168 - 0.985i)T \) |
| 83 | \( 1 + (-0.993 + 0.112i)T \) |
| 89 | \( 1 + (0.970 + 0.239i)T \) |
| 97 | \( 1 + (0.457 - 0.889i)T \) |
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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
−18.11348992688913002400526762091, −17.35298419815219525224451996211, −16.4034722900363832480561272085, −15.73590069147036685741985520608, −15.41195703602368913114770486387, −14.83121960167901900557856859989, −14.454182757943803441110126105953, −13.54964689467939257647421182710, −13.117532351332548867450076611794, −12.24243342363220591093185718931, −11.41687423229215189113745861420, −10.97202236435994501646522464720, −10.40837425925892280468403572646, −9.123014938003500399338219272194, −8.54829111313020116626708430595, −8.21297827495892218259025052951, −7.24999322283088811306714791783, −6.57999164297467136533891969432, −6.00084394715797853208376717663, −4.86884872111621167732905941261, −4.504646323577496544440088184591, −3.6926730507313197717971046814, −3.01065439947354739184098295382, −2.48432638912638130572798140123, −1.726511213047655942687077112397,
0.55123189615188637671397130759, 1.33603645584750485578570703037, 1.83498030727422777627754279977, 2.99601764050996846940306847426, 3.53802443738592302886191205855, 4.32647580301088331166716975905, 4.65701368074612298996065596859, 5.757133314605397295500064303127, 6.60025850998098368701168664202, 7.14412559319643262550556471778, 7.94531745711945914718400871790, 8.570152873055112722951988426627, 9.2197199586620252303382887323, 10.21799356192548625284676979963, 10.83139129432411346777849991902, 11.61110719854856193662438350460, 12.14194687621309617210462534094, 12.99573311656515472829660519972, 13.31529282341664959435194232201, 13.83887124630157244580848780516, 14.5426399628290436134281266836, 15.32235374378975273368952840188, 15.72649978069885456481412222978, 16.51310518012820104598241709495, 17.3540385807598042400087120773