L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.951 − 0.309i)20-s − 23-s + (0.104 − 0.994i)25-s + (0.207 + 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.951 − 0.309i)20-s − 23-s + (0.104 − 0.994i)25-s + (0.207 + 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01882734988 + 0.3924146791i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01882734988 + 0.3924146791i\) |
\(L(1)\) |
\(\approx\) |
\(0.5471531215 + 0.3074341859i\) |
\(L(1)\) |
\(\approx\) |
\(0.5471531215 + 0.3074341859i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.207 + 0.978i)T \) |
| 5 | \( 1 + (-0.743 + 0.669i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (-0.406 - 0.913i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.104 + 0.994i)T \) |
| 31 | \( 1 + (0.207 - 0.978i)T \) |
| 37 | \( 1 + (0.406 - 0.913i)T \) |
| 41 | \( 1 + (-0.587 + 0.809i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.994 + 0.104i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.406 - 0.913i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.743 + 0.669i)T \) |
| 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 | \( 1 + (-0.951 + 0.309i)T \) |
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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
−20.59435262948861012342274168651, −19.853178428634093135596207524302, −19.05112369192182202441801288092, −18.71200503475399571873275447990, −17.745632804212190055111671917498, −16.78267460618603456222885444254, −16.15063647287494683108973570770, −15.354986297109093896173346104585, −14.292549104465813323008813914081, −13.44920795757725556950867719118, −12.51825611968060687229134925782, −12.08210778603039379427470883803, −11.57048469131848379737107359671, −10.35867910701225522466801132778, −9.714419320634530635999057257179, −8.8517390312446349727726794075, −8.24582766746738087344158891219, −7.42452364245019238828546872684, −6.00557828884514005048989538782, −5.159656826562113149569886065982, −4.20631486181426324569911423230, −3.43960166577789896861656874812, −2.53326142668575822290796110984, −1.45393916056977959242123300503, −0.20577846790409098506655042469,
1.044670976333429792759296102619, 2.73824978400580016081334684317, 3.92030139792129670373863908382, 4.26430655474082720372795996651, 5.653251135819345466649721240254, 6.43491315662877379459273964331, 7.190232636913041867579003114273, 7.756290992668977866337622036002, 8.61307934582060958544049652088, 9.64012157803779385601082448499, 10.39633741292392124448840981428, 11.032305771168991628429135257956, 12.2776619669843480051521997983, 13.074887233696017596610293520586, 13.94052074642441217210759294544, 14.61022214319264437279814225158, 15.33365476923351435243720195162, 16.07416635087810207425789194630, 16.686938236152674316768500036381, 17.50934607898686554928879476499, 18.274164831440561329963146571012, 19.1464976570977250772288004018, 19.56462979008365178347697365969, 20.39755250338747128169149912785, 21.8299458662423681337494159791