L(s) = 1 | + (−0.851 − 0.524i)2-s + (−0.913 − 0.406i)3-s + (0.449 + 0.893i)4-s + (0.564 + 0.825i)6-s + (0.0855 − 0.996i)8-s + (0.669 + 0.743i)9-s + (−0.0475 − 0.998i)12-s + (−0.254 − 0.967i)13-s + (−0.595 + 0.803i)16-s + (0.640 + 0.768i)17-s + (−0.179 − 0.983i)18-s + (0.969 − 0.244i)19-s + (−0.580 − 0.814i)23-s + (−0.483 + 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.309 − 0.951i)27-s + ⋯ |
L(s) = 1 | + (−0.851 − 0.524i)2-s + (−0.913 − 0.406i)3-s + (0.449 + 0.893i)4-s + (0.564 + 0.825i)6-s + (0.0855 − 0.996i)8-s + (0.669 + 0.743i)9-s + (−0.0475 − 0.998i)12-s + (−0.254 − 0.967i)13-s + (−0.595 + 0.803i)16-s + (0.640 + 0.768i)17-s + (−0.179 − 0.983i)18-s + (0.969 − 0.244i)19-s + (−0.580 − 0.814i)23-s + (−0.483 + 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.309 − 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3083833420 + 0.1180363008i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3083833420 + 0.1180363008i\) |
\(L(1)\) |
\(\approx\) |
\(0.4723393205 - 0.1840491709i\) |
\(L(1)\) |
\(\approx\) |
\(0.4723393205 - 0.1840491709i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.851 - 0.524i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (-0.254 - 0.967i)T \) |
| 17 | \( 1 + (0.640 + 0.768i)T \) |
| 19 | \( 1 + (0.969 - 0.244i)T \) |
| 23 | \( 1 + (-0.580 - 0.814i)T \) |
| 29 | \( 1 + (0.466 - 0.884i)T \) |
| 31 | \( 1 + (-0.935 - 0.353i)T \) |
| 37 | \( 1 + (-0.988 - 0.151i)T \) |
| 41 | \( 1 + (-0.610 + 0.791i)T \) |
| 43 | \( 1 + (-0.654 + 0.755i)T \) |
| 47 | \( 1 + (0.179 - 0.983i)T \) |
| 53 | \( 1 + (0.595 + 0.803i)T \) |
| 59 | \( 1 + (0.380 - 0.924i)T \) |
| 61 | \( 1 + (-0.879 - 0.475i)T \) |
| 67 | \( 1 + (-0.723 - 0.690i)T \) |
| 71 | \( 1 + (0.897 - 0.441i)T \) |
| 73 | \( 1 + (0.861 + 0.508i)T \) |
| 79 | \( 1 + (-0.123 + 0.992i)T \) |
| 83 | \( 1 + (-0.736 + 0.676i)T \) |
| 89 | \( 1 + (-0.786 - 0.618i)T \) |
| 97 | \( 1 + (-0.516 - 0.856i)T \) |
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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
−18.0110052755973772385901611311, −17.47386086218509402628599110755, −16.610494384639510561063038050078, −16.3071994393856461758837975746, −15.74030621475835572558107811337, −14.96374618661288455459655544716, −14.177153303205590055116088356626, −13.6385557354165525612785774504, −12.205699916990220608592336472, −11.91717799591663165805572822284, −11.18972851431928628690202641107, −10.41483843816252012736962528811, −9.84025881633185645532715891023, −9.25611729765325824742337259376, −8.579740720396553076336046737422, −7.32357693898576981188352811182, −7.17929776380688254071218224878, −6.28185955783950573778138474568, −5.32550771210174078457419703646, −5.19199404008907462409433645684, −4.02482886749158823151462415226, −3.103240760096097811003153105194, −1.789449411229792246455393428449, −1.18376081766081898793695263604, −0.12428322239775070080489876900,
0.581741127593081291833539125828, 1.37766326349483215345607740382, 2.17893180976093032733684767483, 3.10573157216754629263079432342, 3.940020774821915205905855026165, 4.94932515955477341963344381138, 5.73746965130915421752422426044, 6.49455967058814280557800905084, 7.25096705259820960991386552147, 7.954780410285013514690412914, 8.40420290887452194052513544692, 9.63305383093031740450611481411, 10.0881544451337537469913498592, 10.71866938446629649890435462816, 11.386834408918841376115618610716, 12.12615376690379340208107795259, 12.5327344982288147153709169621, 13.22383064607145280209808673876, 14.02953335775215040487259278716, 15.233429222975307474601455547492, 15.727674901717452239958574619, 16.75905537605965959802332895866, 16.8518376855997446367337520947, 17.753817683609229271030662342500, 18.31124418191184690245654957461