L(s) = 1 | + (0.984 − 0.173i)3-s + (−0.642 − 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.342 − 0.939i)13-s + (−0.342 − 0.939i)17-s + (−0.173 − 0.984i)19-s + (−0.766 − 0.642i)21-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.342 + 0.939i)33-s + (0.173 − 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)3-s + (−0.642 − 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.342 − 0.939i)13-s + (−0.342 − 0.939i)17-s + (−0.173 − 0.984i)19-s + (−0.766 − 0.642i)21-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.342 + 0.939i)33-s + (0.173 − 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9062900924 - 1.345290517i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9062900924 - 1.345290517i\) |
\(L(1)\) |
\(\approx\) |
\(1.198365817 - 0.4031406058i\) |
\(L(1)\) |
\(\approx\) |
\(1.198365817 - 0.4031406058i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.984 - 0.173i)T \) |
| 7 | \( 1 + (-0.642 - 0.766i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.342 - 0.939i)T \) |
| 17 | \( 1 + (-0.342 - 0.939i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.342 + 0.939i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−21.15254004343641866258702162474, −20.06558624710078437545035723333, −19.238266395366341155329572777888, −18.865168765270175791646192899023, −18.26316972740435538862326025743, −16.89604521067190383590970504996, −16.27306337248407891659268312254, −15.5581157997085614072927930178, −14.86984815918457644856549713022, −14.157351794427264538565346314415, −13.12563574698103108079200477223, −12.94131218880388037928807420492, −11.70313525188222974786200906312, −10.89275833456252066536150494666, −9.91065889969919956927677667425, −9.25715863653935900846409929794, −8.51218426222985469579574425678, −7.95811145991751082237274755682, −6.78147754916050379575532956176, −6.04170426346343942651892665912, −5.074204472345536618784582008581, −3.838553331358040788738658861096, −3.37001832768475625744435028416, −2.326406858945280284358238786550, −1.543364902249438142488393585905,
0.501454232581264215816974247150, 1.76670686950322226994281659780, 2.831777974113658397949415543365, 3.37730316555615958168053418132, 4.45169723867767325345125645078, 5.23703177687743514504792412606, 6.71393478136221375023892587631, 7.14210642217125341425882684629, 7.878984347754012152457424930, 8.92896449432190636924476139774, 9.48703821097254992345806204085, 10.39060542942777215462748088671, 10.98630912694955679884617360259, 12.39142900321528116888487175578, 13.02087833251307010993219198085, 13.43909054237220359178692138614, 14.34854621661195624211311619449, 15.20880285369539429612695654398, 15.69756076233630027839414962632, 16.574041515670039256051701644525, 17.598354885574055672275485039900, 18.23956189410624880681400237223, 19.00422653583765200698563921553, 19.88730675835828948097526216065, 20.40967475638340890334869813243