L(s) = 1 | − 2-s + (−0.939 + 0.342i)3-s + 4-s + (−0.766 − 0.642i)5-s + (0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s − 8-s + (0.766 − 0.642i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)12-s + (−0.766 − 0.642i)13-s + (−0.766 − 0.642i)14-s + (0.939 + 0.342i)15-s + 16-s − 17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.939 + 0.342i)3-s + 4-s + (−0.766 − 0.642i)5-s + (0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s − 8-s + (0.766 − 0.642i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)12-s + (−0.766 − 0.642i)13-s + (−0.766 − 0.642i)14-s + (0.939 + 0.342i)15-s + 16-s − 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0287 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0287 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3063161789 - 0.2976256270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3063161789 - 0.2976256270i\) |
\(L(1)\) |
\(\approx\) |
\(0.4703910118 + 0.02360871187i\) |
\(L(1)\) |
\(\approx\) |
\(0.4703910118 + 0.02360871187i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.766 - 0.642i)T \) |
| 13 | \( 1 + (-0.766 - 0.642i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.766 - 0.642i)T \) |
| 31 | \( 1 + (-0.173 - 0.984i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.939 + 0.342i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.766 - 0.642i)T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.173 - 0.984i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−18.37337749165793268190581108058, −17.96977466063305701046879608136, −17.348133400906898343442573605163, −16.842274796349269474404522593530, −16.11533199137701845854475239831, −15.425875221180149962923559732636, −14.6611623543411859712736826083, −14.118085870248082407511728582590, −12.822116404406622408789874943074, −12.20314274184972517159374959851, −11.41256831233410356210672895589, −11.1680781462209464428709629583, −10.56980765094673066706676332963, −9.719852113546352171151225362511, −8.9350685354731163495564633800, −8.004365996604084729151161722518, −7.295802960898053284402658679851, −6.80172389244041504571213851196, −6.57187969309985158249901897844, −5.070805833950063233749804981161, −4.566339190671942300372404521578, −3.60199938143789880276203884156, −2.38966766434803092390897719403, −1.70928636197654436944652689712, −0.77812242775102944004686662942,
0.281023728709558089412878251544, 1.2012214834603798561454611996, 1.9649896067241129986932285022, 3.239045090642399273910037266044, 4.037157237896360176385089279920, 4.989555336806438115276431840742, 5.622037351606406429634222285380, 6.28596718406781985140341214527, 7.344126992031117893911615191773, 7.82697472304183921214419426695, 8.65751052167575565626818330024, 9.28934208576212879572299622319, 9.879704120267705607345156258434, 10.94933705885491775412496390272, 11.384718071296466613710963952388, 11.83848065824546704376225024179, 12.40038277563287630105846344421, 13.29088437367703262200254119193, 14.68279213734895592615251718398, 15.249144613265696903976760071033, 15.62271517556229446780175654200, 16.59820765786987394182006715072, 16.85826682100661470057281604217, 17.58149474836072229029792075286, 18.145582003140634008748275057616