L(s) = 1 | + (0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + i·12-s + (−0.951 − 0.309i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + (0.809 − 0.587i)19-s + 21-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + i·12-s + (−0.951 − 0.309i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + (0.809 − 0.587i)19-s + 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.707974118 + 1.770541307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.707974118 + 1.770541307i\) |
\(L(1)\) |
\(\approx\) |
\(2.010509378 + 0.8740521592i\) |
\(L(1)\) |
\(\approx\) |
\(2.010509378 + 0.8740521592i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.587 + 0.809i)T \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 13 | \( 1 + (-0.951 - 0.309i)T \) |
| 17 | \( 1 + (-0.951 + 0.309i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.587 - 0.809i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.587 - 0.809i)T \) |
| 53 | \( 1 + (-0.951 - 0.309i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.587 + 0.809i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.951 - 0.309i)T \) |
| 89 | \( 1 - T \) |
| 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
−32.24239574939017536368366603291, −31.27319866070328329500585726727, −30.7557184708851293766202306484, −29.43954117173196649041801666039, −28.66486936651011508887660184115, −26.968278032074018810557586885171, −25.26563782786901351305311302693, −24.58933108698170895767017536496, −23.67553876730457443048068877961, −22.26284032013705932237979761198, −21.14195925722907328762159607574, −19.98727178602226004683273893887, −18.99680973227963566327431523880, −17.70430554696065935721439631452, −15.666798026076328082553958688348, −14.58529741217483543797274575657, −13.64830915552557132050017113315, −12.31677251168010655885873954675, −11.52983950795412698090652758470, −9.51798462482897505199703962805, −7.82916659086975939655955205908, −6.44015500418437437421037453011, −4.90456119209350543290966291353, −2.97992395557419224278476627286, −1.723758198772869418505426868613,
2.545932049011275066614979518405, 4.12435399560818841842938896144, 5.08425539168836284400349572046, 7.09660451208739003267831874638, 8.373555550569358350458605813068, 10.23553166496727196273848911659, 11.42750582819539110062634746170, 13.169825139446901921591573677410, 14.25604542599703243496958690568, 15.11917739858645865246968138043, 16.33803739145435748865479265172, 17.46038155300414156323913067588, 19.826346326391019866049351022005, 20.460024069548654974207827477952, 21.69579458720232171002468576725, 22.54543636843692352261924722193, 24.007808607575663651442583849727, 24.93362129115886227139948266660, 26.33128059022188721253319186672, 26.98395677187280388269892873256, 28.68536795627635557856742923621, 30.151033722391601601814070160676, 30.9694373845960468966597706857, 32.06824360155896204167259720299, 32.98856084270051751682974877560