L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.913 − 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (−0.104 + 0.994i)11-s + (0.669 + 0.743i)12-s + (0.669 − 0.743i)13-s + (−0.104 − 0.994i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.104 − 0.994i)17-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.913 − 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (−0.104 + 0.994i)11-s + (0.669 + 0.743i)12-s + (0.669 − 0.743i)13-s + (−0.104 − 0.994i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.104 − 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3110370237 - 0.5382880390i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3110370237 - 0.5382880390i\) |
\(L(1)\) |
\(\approx\) |
\(0.6034514472 - 0.5344453494i\) |
\(L(1)\) |
\(\approx\) |
\(0.6034514472 - 0.5344453494i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.913 - 0.406i)T \) |
| 11 | \( 1 + (-0.104 + 0.994i)T \) |
| 13 | \( 1 + (0.669 - 0.743i)T \) |
| 17 | \( 1 + (-0.104 - 0.994i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.809 + 0.587i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.978 + 0.207i)T \) |
| 43 | \( 1 + (0.669 + 0.743i)T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (0.913 + 0.406i)T \) |
| 59 | \( 1 + (-0.978 - 0.207i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.913 + 0.406i)T \) |
| 73 | \( 1 + (-0.104 + 0.994i)T \) |
| 79 | \( 1 + (-0.104 - 0.994i)T \) |
| 83 | \( 1 + (-0.978 + 0.207i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−37.16756181977935957341332185205, −35.33332501874698288304989885068, −34.565921281326361611227689466503, −33.83848465190598835468879863547, −32.68650886972354194059299581543, −31.102289562994732026957308724036, −30.1516804858797758646847114488, −28.25533759355966490337580506695, −27.09122899643455057716353748650, −26.1862412657695435868517349919, −24.20744914362598494459972157253, −23.62209553133215726271550153178, −22.17794764041459807961344269522, −21.45731525884525415567104504034, −18.69203619978809376093431825129, −17.82766498168819010446119747056, −16.34619759159998746027301860486, −15.30050854792731329600796505866, −13.96209449937104000167947936788, −12.00031661497892592790821252126, −10.82352262749934549132098129246, −8.52731738891497678888153118368, −6.87773670743363827984138677727, −5.60098766832234001236873175728, −3.9532594390772642425517817320,
1.32588585234507067359422873172, 4.298850718067870172115084159330, 5.38088687439200949180934465211, 7.8749552570529213699195399357, 9.95160054772889519555023459350, 11.41450205462944942836807675666, 12.27127261348917055068066935334, 13.60905162800547447422009118931, 15.615633056435682783925675261647, 17.37202372725927627512602871463, 18.34506544900951357549914188216, 20.15516949634351737809908176036, 20.95520891211973848839806950683, 22.6793634977704134296817678388, 23.46007769912824361314362507822, 24.60858878317887351630515720342, 27.264579035522765226885331341031, 27.87071150710215217353712949464, 28.93192945063543985914287011636, 30.213571781056725489155528923456, 31.17662863362418395813916760153, 32.71708610605082383893812963499, 33.779419814769545755870531510386, 35.6058569808683897985311616582, 36.24320029802769065796389121025