L(s) = 1 | + (−0.994 − 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.951 − 0.309i)13-s + (0.406 − 0.913i)17-s + (−0.104 − 0.994i)19-s + (−0.743 + 0.669i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.913 + 0.406i)31-s + (0.994 − 0.104i)33-s + (−0.207 + 0.978i)37-s + (−0.978 + 0.207i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.951 − 0.309i)13-s + (0.406 − 0.913i)17-s + (−0.104 − 0.994i)19-s + (−0.743 + 0.669i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.913 + 0.406i)31-s + (0.994 − 0.104i)33-s + (−0.207 + 0.978i)37-s + (−0.978 + 0.207i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01758674187 - 0.2108085281i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01758674187 - 0.2108085281i\) |
\(L(1)\) |
\(\approx\) |
\(0.6939003715 - 0.08742072309i\) |
\(L(1)\) |
\(\approx\) |
\(0.6939003715 - 0.08742072309i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.994 - 0.104i)T \) |
| 11 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (0.951 - 0.309i)T \) |
| 17 | \( 1 + (0.406 - 0.913i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (-0.743 + 0.669i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (-0.207 + 0.978i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.406 + 0.913i)T \) |
| 53 | \( 1 + (-0.994 - 0.104i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.406 + 0.913i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.207 - 0.978i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.587 + 0.809i)T \) |
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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
−21.13623883697666044681164991582, −20.497962407753806103903491875332, −19.30342607092584753496693031792, −18.43603181774930014323291945811, −18.21448582767886904904131618387, −17.11229228112492215851438899572, −16.45829025285327284964377361291, −15.92272013981261650546643870301, −15.07680395140333947279290228606, −14.14608030401786267029333976925, −13.1263884971434644448551610753, −12.59980273333633011441695390638, −11.74429787096939392740286285672, −10.88509637577076105058401312222, −10.394921635103113302457399162579, −9.58663285024772901494041184851, −8.35337868834284841788937529110, −7.77327130875728820707662255384, −6.55889703839697995843712605580, −5.97280589369285917427947161856, −5.26403916113778324249431935452, −4.21301500944278894231831758411, −3.49828389270748560042109464346, −2.07032219510597451618431556898, −1.11708483408278774375294797658,
0.059001346806344997027381018101, 0.94671174876918636257455075888, 2.107116810164280098370936739644, 3.23049020116382196012381403312, 4.34198675626612893343356649630, 5.2138030147107169826435921712, 5.79106322558309964900344400419, 6.77343407804155169051965568099, 7.517044379252705954321575527464, 8.36141530607747529591386106269, 9.5320638122678006609567489181, 10.252644428594181253409698371738, 11.01334344032727016359983533017, 11.679327590442146387053552952631, 12.45588596925045456637175015723, 13.3780794936891516685781699443, 13.76180644025505957076293880104, 15.24980815782407546901655923746, 15.756070494438438199976709174679, 16.30774365377674613808664866682, 17.46529504051736611781609453421, 17.7514739460092869496865690810, 18.67374404422172310855418423866, 19.15067077354547982300785495319, 20.60583266831620761734320616189