L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (−0.994 − 0.104i)19-s + (0.207 − 0.978i)20-s + (−0.5 − 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.743 + 0.669i)28-s + (0.809 − 0.587i)29-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (−0.994 − 0.104i)19-s + (0.207 − 0.978i)20-s + (−0.5 − 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.743 + 0.669i)28-s + (0.809 − 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.088910703 - 2.156866925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.088910703 - 2.156866925i\) |
\(L(1)\) |
\(\approx\) |
\(1.527929534 - 0.8395156647i\) |
\(L(1)\) |
\(\approx\) |
\(1.527929534 - 0.8395156647i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (0.743 - 0.669i)T \) |
| 7 | \( 1 + (-0.994 + 0.104i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.994 - 0.104i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.207 - 0.978i)T \) |
| 37 | \( 1 + (-0.994 + 0.104i)T \) |
| 41 | \( 1 + (0.994 + 0.104i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.994 - 0.104i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.587 - 0.809i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.207 + 0.978i)T \) |
| 73 | \( 1 + (0.587 + 0.809i)T \) |
| 79 | \( 1 + (0.669 - 0.743i)T \) |
| 83 | \( 1 + (0.207 + 0.978i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.207 - 0.978i)T \) |
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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
−21.600680114638088644479954077971, −20.75952805319627010669792808908, −19.69392304019846383385889749335, −19.28799513796965423570138695788, −17.97024948046812631426270600765, −17.46266129601864540042951025653, −16.502480179642791156729732873165, −15.78005931599234132255562599261, −15.14274304367361902499842879010, −14.140073716019931412763378176855, −13.72417598597579571896742086772, −12.87411806426050691077436423835, −12.30152896356692095507432581282, −11.10606467826818725014321221347, −10.55872944597054455569032325129, −9.6106556335925174156481193025, −8.662241152209846679305830388785, −7.47376745263442425269907446202, −6.60887937051908797759105347107, −6.297325370861744740530198043494, −5.325229630986305272900544356274, −4.30815254755634994541914086977, −3.33744213251681126771830169940, −2.65688676342195948673337611822, −1.714134473832546278523539351233,
0.60664196535830108692420693921, 2.06359696949885492499037693310, 2.52821256725251952772644523247, 3.827314542237512315269620524164, 4.511729234914800426815054107413, 5.45651665178868700994451938375, 6.39210550050239516540177374322, 6.63172383605711929790805957940, 8.16780456271651999011152941381, 9.1105740229240490141902741192, 9.95238035648765619803890234036, 10.55663645012201710183479711150, 11.59472483513020180881334884353, 12.59916902030673579042660275229, 12.887354166024252293706266548789, 13.65685883347220451880387738527, 14.37350519895660280209154414055, 15.42975225368741012054505328631, 15.97030600080413017911523050469, 16.80273846219436706208526468727, 17.542777819258831736936744289066, 18.70107074486157984151953147348, 19.46656274688487349715134245589, 20.08110865525734460516480710952, 20.903226603055468324491969114224