L(s) = 1 | + (0.902 − 0.429i)2-s + (0.997 − 0.0634i)3-s + (0.630 − 0.776i)4-s + (−0.786 − 0.618i)5-s + (0.873 − 0.486i)6-s + (−0.916 − 0.400i)7-s + (0.235 − 0.971i)8-s + (0.991 − 0.126i)9-s + (−0.975 − 0.220i)10-s + (0.841 + 0.540i)11-s + (0.580 − 0.814i)12-s + (−0.0792 + 0.996i)13-s + (−0.999 + 0.0317i)14-s + (−0.823 − 0.567i)15-s + (−0.204 − 0.978i)16-s + (0.0475 − 0.998i)17-s + ⋯ |
L(s) = 1 | + (0.902 − 0.429i)2-s + (0.997 − 0.0634i)3-s + (0.630 − 0.776i)4-s + (−0.786 − 0.618i)5-s + (0.873 − 0.486i)6-s + (−0.916 − 0.400i)7-s + (0.235 − 0.971i)8-s + (0.991 − 0.126i)9-s + (−0.975 − 0.220i)10-s + (0.841 + 0.540i)11-s + (0.580 − 0.814i)12-s + (−0.0792 + 0.996i)13-s + (−0.999 + 0.0317i)14-s + (−0.823 − 0.567i)15-s + (−0.204 − 0.978i)16-s + (0.0475 − 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.771585541 - 1.390127615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.771585541 - 1.390127615i\) |
\(L(1)\) |
\(\approx\) |
\(1.752455288 - 0.8299787075i\) |
\(L(1)\) |
\(\approx\) |
\(1.752455288 - 0.8299787075i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (0.902 - 0.429i)T \) |
| 3 | \( 1 + (0.997 - 0.0634i)T \) |
| 5 | \( 1 + (-0.786 - 0.618i)T \) |
| 7 | \( 1 + (-0.916 - 0.400i)T \) |
| 11 | \( 1 + (0.841 + 0.540i)T \) |
| 13 | \( 1 + (-0.0792 + 0.996i)T \) |
| 17 | \( 1 + (0.0475 - 0.998i)T \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.356 - 0.934i)T \) |
| 29 | \( 1 + (-0.266 + 0.963i)T \) |
| 31 | \( 1 + (-0.553 + 0.832i)T \) |
| 37 | \( 1 + (0.766 - 0.642i)T \) |
| 41 | \( 1 + (-0.386 + 0.922i)T \) |
| 43 | \( 1 + (0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.110 + 0.993i)T \) |
| 53 | \( 1 + (-0.823 + 0.567i)T \) |
| 59 | \( 1 + (0.981 + 0.189i)T \) |
| 61 | \( 1 + (-0.142 - 0.989i)T \) |
| 67 | \( 1 + (0.928 - 0.371i)T \) |
| 71 | \( 1 + (-0.999 - 0.0317i)T \) |
| 73 | \( 1 + (-0.987 + 0.158i)T \) |
| 79 | \( 1 + (-0.745 + 0.666i)T \) |
| 83 | \( 1 + (0.580 + 0.814i)T \) |
| 89 | \( 1 + (-0.701 - 0.712i)T \) |
| 97 | \( 1 + (-0.444 - 0.895i)T \) |
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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
−26.84502790195493129299958223848, −25.827382304561182464605162763599, −25.381535983316742323737886221464, −24.29513252159009927197914937069, −23.373145802443385806385690280101, −22.24691658540314489718819919207, −21.78853230963788249280699746420, −20.456484745791705370670435811666, −19.486837042820409836388014673878, −18.99545097141181920939517203027, −17.2652988776832451993270558975, −16.03832636856913810256511967399, −15.14932348366941339603906488225, −14.82591263497567599595573333098, −13.46500825812988277027047720562, −12.76771661746329902808720131760, −11.61352721183604649495256767501, −10.35503548283265840220882395935, −8.84880458931945326520966887251, −7.91158299210833021514559350904, −6.88581632495041205924087602722, −5.872556376993270716562904165573, −4.02352760757681763418482388138, −3.46150354760066724318971797559, −2.41552535701460553025006912137,
1.4021080617512821882519498169, 2.87644281693909305664449653708, 3.98946997058875145840635365517, 4.5926870836219908753048882399, 6.58312515852830287347252498905, 7.31479185895284324674425873947, 8.955330307484869704678999943, 9.695671072310069918932900788687, 11.1115173605932854553154612431, 12.43605356408932907755336585225, 12.8238619724180073452999918323, 14.12316295053067046579066295, 14.74922848400262497363247761559, 15.979563153500889262894350079301, 16.56195311740958294587995959337, 18.713688996653758803814632146389, 19.46771279119567783305197346129, 20.0892474540974593372945963068, 20.80610691600655081239890537372, 21.94375576020879795312291751398, 23.03267225247306744593854387590, 23.75909460416024609636248338090, 24.81313100826560755304641903703, 25.46775511780867991521366475492, 26.75472110448609527685079087879