L(s) = 1 | + (−0.0792 − 0.996i)2-s + (−0.444 − 0.895i)3-s + (−0.987 + 0.158i)4-s + (−0.786 + 0.618i)5-s + (−0.857 + 0.513i)6-s + (0.805 + 0.592i)7-s + (0.235 + 0.971i)8-s + (−0.605 + 0.795i)9-s + (0.678 + 0.734i)10-s + (0.841 − 0.540i)11-s + (0.580 + 0.814i)12-s + (−0.823 + 0.567i)13-s + (0.527 − 0.849i)14-s + (0.902 + 0.429i)15-s + (0.950 − 0.312i)16-s + (0.0475 + 0.998i)17-s + ⋯ |
L(s) = 1 | + (−0.0792 − 0.996i)2-s + (−0.444 − 0.895i)3-s + (−0.987 + 0.158i)4-s + (−0.786 + 0.618i)5-s + (−0.857 + 0.513i)6-s + (0.805 + 0.592i)7-s + (0.235 + 0.971i)8-s + (−0.605 + 0.795i)9-s + (0.678 + 0.734i)10-s + (0.841 − 0.540i)11-s + (0.580 + 0.814i)12-s + (−0.823 + 0.567i)13-s + (0.527 − 0.849i)14-s + (0.902 + 0.429i)15-s + (0.950 − 0.312i)16-s + (0.0475 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7086742668 - 0.1614002511i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7086742668 - 0.1614002511i\) |
\(L(1)\) |
\(\approx\) |
\(0.6933837436 - 0.2910196252i\) |
\(L(1)\) |
\(\approx\) |
\(0.6933837436 - 0.2910196252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (-0.0792 - 0.996i)T \) |
| 3 | \( 1 + (-0.444 - 0.895i)T \) |
| 5 | \( 1 + (-0.786 + 0.618i)T \) |
| 7 | \( 1 + (0.805 + 0.592i)T \) |
| 11 | \( 1 + (0.841 - 0.540i)T \) |
| 13 | \( 1 + (-0.823 + 0.567i)T \) |
| 17 | \( 1 + (0.0475 + 0.998i)T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.630 - 0.776i)T \) |
| 29 | \( 1 + (-0.701 + 0.712i)T \) |
| 31 | \( 1 + (0.997 - 0.0634i)T \) |
| 37 | \( 1 + (-0.939 + 0.342i)T \) |
| 41 | \( 1 + (0.991 + 0.126i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.916 + 0.400i)T \) |
| 53 | \( 1 + (0.902 - 0.429i)T \) |
| 59 | \( 1 + (0.981 - 0.189i)T \) |
| 61 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.928 + 0.371i)T \) |
| 71 | \( 1 + (0.527 + 0.849i)T \) |
| 73 | \( 1 + (0.356 + 0.934i)T \) |
| 79 | \( 1 + (-0.204 + 0.978i)T \) |
| 83 | \( 1 + (0.580 - 0.814i)T \) |
| 89 | \( 1 + (0.967 + 0.251i)T \) |
| 97 | \( 1 + (-0.553 - 0.832i)T \) |
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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
−27.04297917942600815213806451818, −26.326172796668237216934317004385, −24.86711176405244429801913881833, −24.26186680955881776695084066872, −23.075791791515687120039692787336, −22.70927492643090213610226194922, −21.430659672697760217018519762930, −20.32240847562497574482613551683, −19.47167706141734441302143552557, −17.72650590551076792323658365117, −17.2538514673768931920531268659, −16.397290950896154998067840909427, −15.356537717439908034171067924591, −14.831913244853415742108450813568, −13.60108786899536727288671547274, −12.14547385366648332024556396443, −11.231218663429620330492914751633, −9.81155238514931898949286197619, −9.021505090334016024594649332146, −7.79238970608917011005519521613, −6.89725840869827329853400970687, −5.147268380289672694557306037556, −4.742256947111646443733688447202, −3.66709245593512768531704599954, −0.71124096845744992460112946648,
1.36610535589662192830175762303, 2.52703524452067291659609852531, 3.92448857841215943156109988043, 5.24905257689743763031176764408, 6.645046993835733684715898605125, 7.96637348483430118200377729877, 8.69965340620397854883153170729, 10.37970579475667504286054704630, 11.41288758472376587374076977853, 11.882624427503019701026754095367, 12.731443300962508938989667688904, 14.24260755031885774878475851545, 14.69882740885809840628751879134, 16.65254233679422028113834799336, 17.51786295217385141586408381187, 18.6373122756243992009024532161, 19.04674789256813235585564987663, 19.87133683737857188346457585855, 21.25024756983056017191082412289, 22.19759987180603513548441280697, 22.85662958853164951126158142836, 23.971082668443397954719539063733, 24.64351521925089971436249536002, 26.17411377595378049074461509705, 27.21258368064203314901387810800