L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 13-s + 15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 13-s + 15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06601812957 - 0.2818140891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06601812957 - 0.2818140891i\) |
\(L(1)\) |
\(\approx\) |
\(0.6247622464 - 0.1157391676i\) |
\(L(1)\) |
\(\approx\) |
\(0.6247622464 - 0.1157391676i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.85853517182838280416249869302, −22.987778812485892115156221745116, −21.85865383117435078741876379745, −21.61767058443347370133732592073, −20.42431788746501810071108792238, −19.807552216497834144163074712656, −18.949489205508623654729575213515, −17.54432656803858569110150422311, −16.907950871471791714954712833448, −16.36407121089021409743607370217, −15.349786777855022348324229510950, −14.77020137808521568767752161522, −13.48283884003268687942700974791, −12.48078594257674426880802162518, −11.65237655139595589310231866661, −10.984566540471021701740718291060, −9.83805371164574265657578413696, −9.03141103763543614997936236152, −8.27286398555495895685304155005, −6.97041600874877999706710382532, −5.769599144285538908346265218903, −4.99879852387997315658066060865, −4.07703197266772149901781481375, −3.22286924142962025370033226933, −1.38841303424923752295736507827,
0.165477521250542596541130516220, 1.93921381223955649410864974061, 2.75952467154092144822792860785, 4.21210568457401116080116418328, 5.247473960804486451098068722456, 6.51342251203297494308737096472, 7.15209667511041768346307437988, 7.73370789808867938771995427077, 9.10966253067953668845571781396, 10.221605094339495197653422134520, 11.19175815340865036694658628118, 11.87800825717086168375886388449, 12.62338404580540089823875389781, 13.65947310872802176098447220583, 14.63889267683166509422234978915, 15.237671070700496049573186588914, 16.578762732185347127246949537948, 17.23181884647965897399633728887, 18.29637495946304937614486221421, 18.6378079571581209297564447280, 19.76731992766924095999498729720, 20.21123832252171398134211597510, 21.80291075783481160357568573189, 22.541721929101795104651249143289, 22.92749312241931402857201037173