L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + 7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (−0.5 − 0.866i)29-s − 31-s + (0.5 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + 7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (−0.5 − 0.866i)29-s − 31-s + (0.5 − 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6702831590 - 0.7393336329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6702831590 - 0.7393336329i\) |
\(L(1)\) |
\(\approx\) |
\(0.8784852387 - 0.08627270793i\) |
\(L(1)\) |
\(\approx\) |
\(0.8784852387 - 0.08627270793i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−28.21300298767405599784603495293, −27.0350567918198331997344419043, −26.05318154751820883650400859799, −25.019649961628795226401989110024, −24.02705379396665348118788829277, −23.3536352158540994988740311351, −22.135597982439202327457333437042, −21.38826951733716546126248662252, −20.05186417128156877020876717678, −18.72403791186959078441133711206, −18.13743811806040214757061947144, −17.42395661087770732086658788153, −16.14129526409035133335893654100, −14.63789659037557292210955200821, −13.90099363797751997015944483973, −12.85116071379159551357267051154, −11.43021572382668854596420530275, −10.968010947447759195067924612, −9.486298687036911477874182228208, −7.8285565242584904344911801114, −7.13373696189748915811067560762, −5.866082221427460626793246200764, −4.80773770484773524438272910154, −2.66164525598229018735477818742, −1.66345623240916865804959961507,
0.38265996194905344818226957430, 2.21891303244719861833974789665, 4.152745591535586326720891717137, 5.12958050884839852194235039923, 5.87459184385586371997939686829, 7.86903995391234338848912983671, 8.84924496923572368691635071743, 10.11931507413839285510109711777, 10.87873362700157441945969091570, 12.1802747400484667116197414974, 13.17432663969937261120643316902, 14.61389039005498075024599960797, 15.48545303141122063154256537242, 16.595449427373586071527635737618, 17.457074808352611804473731913463, 18.15892642534942719142997136472, 20.04231179222773193641543487647, 20.77172412105112433518634813188, 21.48495231367675016982623508617, 22.455025535917180697928606231743, 23.770944389785628160334843677970, 24.38333077692188520478889293701, 25.68622429885691521355593399258, 26.731161797196966770617689823027, 27.645893230036051767835912625124