L(s) = 1 | − i·2-s + (0.686 + 0.727i)3-s − 4-s + (−0.549 − 0.835i)5-s + (0.727 − 0.686i)6-s + (−0.0581 − 0.998i)7-s + i·8-s + (−0.0581 + 0.998i)9-s + (−0.835 + 0.549i)10-s + (0.835 + 0.549i)11-s + (−0.686 − 0.727i)12-s + (−0.549 − 0.835i)13-s + (−0.998 + 0.0581i)14-s + (0.230 − 0.973i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯ |
L(s) = 1 | − i·2-s + (0.686 + 0.727i)3-s − 4-s + (−0.549 − 0.835i)5-s + (0.727 − 0.686i)6-s + (−0.0581 − 0.998i)7-s + i·8-s + (−0.0581 + 0.998i)9-s + (−0.835 + 0.549i)10-s + (0.835 + 0.549i)11-s + (−0.686 − 0.727i)12-s + (−0.549 − 0.835i)13-s + (−0.998 + 0.0581i)14-s + (0.230 − 0.973i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.057468420 - 0.6596044271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.057468420 - 0.6596044271i\) |
\(L(1)\) |
\(\approx\) |
\(0.8179233968 - 0.3964704963i\) |
\(L(1)\) |
\(\approx\) |
\(0.8179233968 - 0.3964704963i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.686 + 0.727i)T \) |
| 5 | \( 1 + (-0.549 - 0.835i)T \) |
| 7 | \( 1 + (-0.0581 - 0.998i)T \) |
| 11 | \( 1 + (0.835 + 0.549i)T \) |
| 13 | \( 1 + (-0.549 - 0.835i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.642 + 0.766i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (-0.998 + 0.0581i)T \) |
| 31 | \( 1 + (-0.918 - 0.396i)T \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.686 + 0.727i)T \) |
| 53 | \( 1 + (-0.286 + 0.957i)T \) |
| 59 | \( 1 + (-0.230 + 0.973i)T \) |
| 61 | \( 1 + (-0.116 + 0.993i)T \) |
| 67 | \( 1 + (0.686 - 0.727i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.0581 - 0.998i)T \) |
| 79 | \( 1 + (-0.230 + 0.973i)T \) |
| 83 | \( 1 + (0.597 - 0.802i)T \) |
| 89 | \( 1 + (0.448 - 0.893i)T \) |
| 97 | \( 1 + (-0.918 + 0.396i)T \) |
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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
−18.46474479696234787734545162093, −17.725869730924612759775062744744, −17.23798906763136109129987886551, −16.19436989178780864642090346671, −15.487170872192938712449512826214, −15.000940596852212018033447646438, −14.42031831932015567041785176617, −13.960592389841604373665480633565, −13.04917287020963943387197695645, −12.46756514467255215715239680220, −11.61518674131549651474989017484, −11.05087896473449648081536847348, −9.64928545119205201903218157615, −9.14846093712258885693711420735, −8.603397316798880561083265860, −7.88904907341040877472819294256, −7.09329027417606749345552188624, −6.61051557003878912892524335786, −6.09255888048489824186669736443, −5.10803585509580666274948469668, −3.8716288393142551821662780560, −3.64022384541462800948750969539, −2.42819540818898598213360691790, −1.79565131537538396821996527874, −0.35703949064487877390585767782,
0.40261947831441549009856533163, 1.41116482497976391528534428239, 2.22233385537600813121642905539, 3.16936101115617739599643447195, 4.01487888034977016687646989816, 4.324545110291148219404461081521, 4.88155898854428837340514433986, 5.94799102314912516180588964742, 7.38035708937438830510919890478, 7.867726371152189051713847264760, 8.63776218314819644932700052466, 9.4091711941962938533260990319, 9.71549396409831738394289883793, 10.73046103725794672975940705204, 11.03196000120800101826484303874, 12.05079119480502841748434561088, 12.80350977556739562450567802231, 13.15079038711178525625410687423, 14.13669284723172984042132860416, 14.63489358666655725524579553510, 15.2768900577428577402702792909, 16.35190310048894821585632939149, 16.78586321870385585883839454637, 17.420514430861721309958466554245, 18.29729061159425794115876974040