L(s) = 1 | + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (0.0682 + 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (−0.962 − 0.269i)10-s + (−0.460 − 0.887i)11-s + (−0.775 + 0.631i)12-s + (−0.682 − 0.730i)13-s + (0.962 − 0.269i)14-s + (0.990 + 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯ |
L(s) = 1 | + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (0.0682 + 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (−0.962 − 0.269i)10-s + (−0.460 − 0.887i)11-s + (−0.775 + 0.631i)12-s + (−0.682 − 0.730i)13-s + (0.962 − 0.269i)14-s + (0.990 + 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5112255252 - 0.5129798991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5112255252 - 0.5129798991i\) |
\(L(1)\) |
\(\approx\) |
\(0.7207814736 - 0.07413702400i\) |
\(L(1)\) |
\(\approx\) |
\(0.7207814736 - 0.07413702400i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (-0.334 + 0.942i)T \) |
| 3 | \( 1 + (0.203 - 0.979i)T \) |
| 5 | \( 1 + (0.0682 + 0.997i)T \) |
| 7 | \( 1 + (-0.576 - 0.816i)T \) |
| 11 | \( 1 + (-0.460 - 0.887i)T \) |
| 13 | \( 1 + (-0.682 - 0.730i)T \) |
| 17 | \( 1 + (0.460 - 0.887i)T \) |
| 19 | \( 1 + (0.0682 - 0.997i)T \) |
| 23 | \( 1 + (0.334 + 0.942i)T \) |
| 29 | \( 1 + (-0.682 + 0.730i)T \) |
| 31 | \( 1 + (-0.203 - 0.979i)T \) |
| 37 | \( 1 + (0.962 + 0.269i)T \) |
| 41 | \( 1 + (-0.854 - 0.519i)T \) |
| 43 | \( 1 + (0.775 + 0.631i)T \) |
| 53 | \( 1 + (0.854 + 0.519i)T \) |
| 59 | \( 1 + (-0.775 + 0.631i)T \) |
| 61 | \( 1 + (0.962 - 0.269i)T \) |
| 67 | \( 1 + (0.576 - 0.816i)T \) |
| 71 | \( 1 + (-0.334 - 0.942i)T \) |
| 73 | \( 1 + (0.917 - 0.398i)T \) |
| 79 | \( 1 + (-0.990 - 0.136i)T \) |
| 83 | \( 1 + (0.460 + 0.887i)T \) |
| 89 | \( 1 + (-0.0682 - 0.997i)T \) |
| 97 | \( 1 + (0.203 - 0.979i)T \) |
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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
−34.08624426938371373977583508640, −32.494586043676924869641614457020, −31.7543500082566136521787736245, −30.8760303356671004519803305332, −28.842038487735578743040404416335, −28.493262642221711797698961549567, −27.41182181385636860491044673765, −26.18147925436510848069086642027, −25.12021698896025453042792028592, −23.11214887243879382827799685167, −21.85376862021384427736362761987, −21.013620330771930889003970785486, −20.05496227208771461412215748353, −18.883731548369477727429158448376, −17.17100814363967066444468045506, −16.23500526937755633770360330317, −14.66074390748963159771247400730, −12.860923768219952941004898228879, −11.95745500345811131378718704460, −10.169079552928959094373792383376, −9.35707945922057366617126212828, −8.25226044793790325538495908855, −5.28613800400254259189317738568, −4.024590165014081591207689955353, −2.24087163137718549623073921964,
0.45984567522448413171395804049, 3.089906704886672584401810120004, 5.69540850183203570520634103562, 7.037118991779392741187685978571, 7.70737113861237291413284923073, 9.53142726175145847411160783060, 11.01849897239731616743764791781, 13.1978988173697519568856364752, 13.96316265180131064052285979369, 15.23561449430988963221831856964, 16.807270715851664141777029488080, 17.968691467682343402058373067232, 18.909015963809700832186936025815, 19.87993576960159633619044717815, 22.24491668338077545303618098927, 23.224509746474363997155444675729, 24.177445593763341101597355538646, 25.48428351998209918131959118568, 26.23662320195752934810110629938, 27.27469353898938894395766136550, 29.191684350514632345362757582796, 29.8733980804565949012937509562, 31.36761999168637493154010584785, 32.41023284522964143665078397151, 33.85603886903175865861188336115