L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.540 − 0.841i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (0.281 + 0.959i)6-s + (−0.755 + 0.654i)7-s + (−0.654 − 0.755i)8-s + (−0.415 + 0.909i)9-s + (−0.415 − 0.909i)10-s + (−0.654 + 0.755i)11-s − i·12-s + (0.540 + 0.841i)13-s + (0.909 − 0.415i)14-s + (0.281 − 0.959i)15-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)17-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.540 − 0.841i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (0.281 + 0.959i)6-s + (−0.755 + 0.654i)7-s + (−0.654 − 0.755i)8-s + (−0.415 + 0.909i)9-s + (−0.415 − 0.909i)10-s + (−0.654 + 0.755i)11-s − i·12-s + (0.540 + 0.841i)13-s + (0.909 − 0.415i)14-s + (0.281 − 0.959i)15-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4950154381 + 0.1551951723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4950154381 + 0.1551951723i\) |
\(L(1)\) |
\(\approx\) |
\(0.5992546385 + 0.01634883912i\) |
\(L(1)\) |
\(\approx\) |
\(0.5992546385 + 0.01634883912i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.755 + 0.654i)T \) |
| 11 | \( 1 + (-0.654 + 0.755i)T \) |
| 13 | \( 1 + (0.540 + 0.841i)T \) |
| 17 | \( 1 + (0.959 - 0.281i)T \) |
| 19 | \( 1 + (0.909 + 0.415i)T \) |
| 23 | \( 1 + (-0.909 - 0.415i)T \) |
| 29 | \( 1 + (0.755 - 0.654i)T \) |
| 31 | \( 1 + (-0.909 + 0.415i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.540 - 0.841i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 + (-0.841 - 0.540i)T \) |
| 53 | \( 1 + (-0.841 + 0.540i)T \) |
| 59 | \( 1 + (-0.540 + 0.841i)T \) |
| 61 | \( 1 + (-0.989 + 0.142i)T \) |
| 67 | \( 1 + (0.841 - 0.540i)T \) |
| 71 | \( 1 + (0.654 - 0.755i)T \) |
| 73 | \( 1 + (0.415 + 0.909i)T \) |
| 79 | \( 1 + (-0.415 - 0.909i)T \) |
| 83 | \( 1 + (0.281 + 0.959i)T \) |
| 97 | \( 1 + (-0.654 - 0.755i)T \) |
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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
−29.78617355714973116388765200738, −29.08725086084046674185073204993, −28.241315742889274861384840092695, −27.362750273719649200113324832409, −26.2117736170017675719766168291, −25.569075148733380799046251511627, −24.127862565906414871170466838603, −23.189948045016227533599914043509, −21.69039724051710694401845587374, −20.65472244650618671539934811066, −19.85483173468467913927159487369, −18.22060954211923798936383956645, −17.31586122623710956364215703836, −16.206223699829315362647194394580, −15.932574971438514498949172236302, −14.1281015842017372059490167382, −12.61773609089414291031135207473, −11.00825182040328371483934897327, −10.09726385108738910612843355018, −9.291492136803960982676577673681, −7.9289705520401758674731059362, −6.13056335969247341631918601468, −5.36216306222168502458198472189, −3.29045229056601040967992385923, −0.826925468539966862104423331719,
1.77677006470218510785723770909, 2.9264577800547961865205592927, 5.80073145394823440523726805245, 6.73080154517392605686858432158, 7.83187287208858716907074302420, 9.47377560936466196660984867192, 10.41864329388023072689322049991, 11.74229577661865972118572103909, 12.62373316469645112876139659374, 14.01533855190995028012254082839, 15.77023042048105783246239722730, 16.792274863686596790890305299831, 18.25979933517605133234690117885, 18.33909314657266827649201928795, 19.47313047743710201629216863293, 20.954455759598103590402393072922, 22.10642409884919586338356025122, 23.13890143938449393762650352557, 24.632791808534736060154293988522, 25.621652327518771122527802933312, 26.11947855241695188855446488697, 27.76599792900863761644156040833, 28.85616895445999797512796856932, 29.13312576782505744721762473205, 30.39146382299904019454523418566