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
−30.39146382299904019454523418566, −29.13312576782505744721762473205, −28.85616895445999797512796856932, −27.76599792900863761644156040833, −26.11947855241695188855446488697, −25.621652327518771122527802933312, −24.632791808534736060154293988522, −23.13890143938449393762650352557, −22.10642409884919586338356025122, −20.954455759598103590402393072922, −19.47313047743710201629216863293, −18.33909314657266827649201928795, −18.25979933517605133234690117885, −16.792274863686596790890305299831, −15.77023042048105783246239722730, −14.01533855190995028012254082839, −12.62373316469645112876139659374, −11.74229577661865972118572103909, −10.41864329388023072689322049991, −9.47377560936466196660984867192, −7.83187287208858716907074302420, −6.73080154517392605686858432158, −5.80073145394823440523726805245, −2.9264577800547961865205592927, −1.77677006470218510785723770909,
0.826925468539966862104423331719, 3.29045229056601040967992385923, 5.36216306222168502458198472189, 6.13056335969247341631918601468, 7.9289705520401758674731059362, 9.291492136803960982676577673681, 10.09726385108738910612843355018, 11.00825182040328371483934897327, 12.61773609089414291031135207473, 14.1281015842017372059490167382, 15.932574971438514498949172236302, 16.206223699829315362647194394580, 17.31586122623710956364215703836, 18.22060954211923798936383956645, 19.85483173468467913927159487369, 20.65472244650618671539934811066, 21.69039724051710694401845587374, 23.189948045016227533599914043509, 24.127862565906414871170466838603, 25.569075148733380799046251511627, 26.2117736170017675719766168291, 27.362750273719649200113324832409, 28.241315742889274861384840092695, 29.08725086084046674185073204993, 29.78617355714973116388765200738