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
−27.21258368064203314901387810800, −26.17411377595378049074461509705, −24.64351521925089971436249536002, −23.971082668443397954719539063733, −22.85662958853164951126158142836, −22.19759987180603513548441280697, −21.25024756983056017191082412289, −19.87133683737857188346457585855, −19.04674789256813235585564987663, −18.6373122756243992009024532161, −17.51786295217385141586408381187, −16.65254233679422028113834799336, −14.69882740885809840628751879134, −14.24260755031885774878475851545, −12.731443300962508938989667688904, −11.882624427503019701026754095367, −11.41288758472376587374076977853, −10.37970579475667504286054704630, −8.69965340620397854883153170729, −7.96637348483430118200377729877, −6.645046993835733684715898605125, −5.24905257689743763031176764408, −3.92448857841215943156109988043, −2.52703524452067291659609852531, −1.36610535589662192830175762303,
0.71124096845744992460112946648, 3.66709245593512768531704599954, 4.742256947111646443733688447202, 5.147268380289672694557306037556, 6.89725840869827329853400970687, 7.79238970608917011005519521613, 9.021505090334016024594649332146, 9.81155238514931898949286197619, 11.231218663429620330492914751633, 12.14547385366648332024556396443, 13.60108786899536727288671547274, 14.831913244853415742108450813568, 15.356537717439908034171067924591, 16.397290950896154998067840909427, 17.2538514673768931920531268659, 17.72650590551076792323658365117, 19.47167706141734441302143552557, 20.32240847562497574482613551683, 21.430659672697760217018519762930, 22.70927492643090213610226194922, 23.075791791515687120039692787336, 24.26186680955881776695084066872, 24.86711176405244429801913881833, 26.326172796668237216934317004385, 27.04297917942600815213806451818