L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.281 + 0.959i)3-s + (−0.959 + 0.281i)4-s + (−0.415 − 0.909i)5-s + (0.989 + 0.142i)6-s + (0.909 − 0.415i)7-s + (0.415 + 0.909i)8-s + (−0.841 − 0.540i)9-s + (−0.841 + 0.540i)10-s + (0.415 − 0.909i)11-s − i·12-s + (0.281 − 0.959i)13-s + (−0.540 − 0.841i)14-s + (0.989 − 0.142i)15-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)17-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.281 + 0.959i)3-s + (−0.959 + 0.281i)4-s + (−0.415 − 0.909i)5-s + (0.989 + 0.142i)6-s + (0.909 − 0.415i)7-s + (0.415 + 0.909i)8-s + (−0.841 − 0.540i)9-s + (−0.841 + 0.540i)10-s + (0.415 − 0.909i)11-s − i·12-s + (0.281 − 0.959i)13-s + (−0.540 − 0.841i)14-s + (0.989 − 0.142i)15-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5091910925 - 0.5367682566i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5091910925 - 0.5367682566i\) |
\(L(1)\) |
\(\approx\) |
\(0.7200886756 - 0.3822748252i\) |
\(L(1)\) |
\(\approx\) |
\(0.7200886756 - 0.3822748252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.281 + 0.959i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.909 - 0.415i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (0.281 - 0.959i)T \) |
| 17 | \( 1 + (0.142 - 0.989i)T \) |
| 19 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (0.540 - 0.841i)T \) |
| 29 | \( 1 + (-0.909 + 0.415i)T \) |
| 31 | \( 1 + (0.540 + 0.841i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.281 + 0.959i)T \) |
| 43 | \( 1 + (-0.909 - 0.415i)T \) |
| 47 | \( 1 + (0.959 - 0.281i)T \) |
| 53 | \( 1 + (0.959 + 0.281i)T \) |
| 59 | \( 1 + (-0.281 - 0.959i)T \) |
| 61 | \( 1 + (-0.755 + 0.654i)T \) |
| 67 | \( 1 + (-0.959 - 0.281i)T \) |
| 71 | \( 1 + (-0.415 + 0.909i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.841 + 0.540i)T \) |
| 83 | \( 1 + (0.989 + 0.142i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−30.78989484810711418183521312295, −30.14369905711646592700784875526, −28.346799136254785754275778177713, −27.70693273169837326198701979561, −26.274835027942461134948691858896, −25.52673973180029606690604791494, −24.33601836479218366175661930541, −23.60045641902588611124636080301, −22.739101055124861316289042788409, −21.58637952986511605290806820562, −19.49533866224676921149850768412, −18.73971985256432308326123971647, −17.75734178916576974002827833254, −17.03652696919004133171865533232, −15.2756078733742466865222673887, −14.60635821618650170412173769346, −13.455953880005420076905374688357, −12.00868010975480502120792550476, −10.8893675214762232730606700697, −9.027060574241981783981131055432, −7.75674333347310237620895394388, −6.95716262026204558265967059100, −5.85941731525629692957690586982, −4.262012000810423030539523183655, −1.896458691234406766849769227362,
0.96953527722124722383002324959, 3.31725364530825656387875220739, 4.46735835379596856840948530473, 5.40745397996761285550395862701, 8.15573362007295431537206039602, 8.91561478956014316883880491213, 10.33269120903745439252265969087, 11.251182531560792556895433163737, 12.17087430674064709244863046425, 13.630630730736198275360489653934, 14.86993430330217852643727171264, 16.45962053433254189612514761521, 17.136721828885173760633772499656, 18.46347662275327746670746649917, 20.01209812112895096452763610797, 20.63238389864773619834122390144, 21.38695487543621732792735165376, 22.656388013337234577699197183225, 23.543464395375491679593376641939, 24.98429897932421554324954885759, 26.7869367390454501502291042323, 27.31288443742184766617170157821, 27.95889838139846656138259903795, 29.08510831939896434751418074663, 30.10704962091435860151537924539