L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + i·8-s + 10-s + 11-s + (−0.866 + 0.5i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (0.866 + 0.5i)22-s + i·23-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + i·8-s + 10-s + 11-s + (−0.866 + 0.5i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (0.866 + 0.5i)22-s + i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.679433848 + 0.4790412145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.679433848 + 0.4790412145i\) |
\(L(1)\) |
\(\approx\) |
\(1.634442590 + 0.3648137235i\) |
\(L(1)\) |
\(\approx\) |
\(1.634442590 + 0.3648137235i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - iT \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + (0.866 - 0.5i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.866 + 0.5i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 - iT \) |
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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
−29.53941121081478701088821088050, −28.62164514574065260557221949504, −27.65737887429245217872938988674, −26.091194528231615993489339919617, −25.00047071575346528428282577313, −24.38423263340127142903662620731, −22.76557554550140793629562453603, −22.0541727546135464850093573687, −21.54225376117293652291576229807, −20.0682592016487303740084605605, −19.20170466381271633941606670435, −18.06157721189284136428953668553, −16.73022775275502791485685190380, −15.11692656280181870825457236550, −14.62418391517676916162504263225, −13.24213308596741424777624301519, −12.45277917716385175725932307038, −11.17448103393858545262667326307, −10.039412044447869220575978732, −9.01709993961204549518800063156, −6.7167422148696059151707088052, −6.03817502938441125216215586004, −4.64769483312447963525925499354, −2.99686485153590288200444529335, −2.00695378719625515366629070085,
2.02888256713343717449494027660, 3.796885544244193839384639550290, 4.89561653577455829271327951865, 6.30470991268706743414012986898, 7.12599313018977199783229573669, 8.79274157560223888525715178803, 10.00818236517034286351878999255, 11.60423209245806260478767261035, 12.76703895992764454173185007195, 13.70340909418721439428609989998, 14.449766727631962949097192083698, 15.9092795585728910229416669022, 17.02545396231443631354075433891, 17.397278084325571806202920189318, 19.47508856859472728762545722779, 20.429760718742288139283284876105, 21.53560511806069788691422555896, 22.33629139881068269771922187760, 23.4181564656842606992401446411, 24.488003937499717572361341282089, 25.22079509415291971071835559617, 26.21654381121120642353367301335, 27.29183508717116298670693221953, 29.05675713406851541770371978969, 29.50732445492732932802855992773