L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.396 − 0.918i)3-s + (0.939 − 0.342i)4-s + (−0.230 − 0.973i)5-s + (−0.549 − 0.835i)6-s + (−0.286 + 0.957i)7-s + (0.866 − 0.5i)8-s + (−0.686 + 0.727i)9-s + (−0.396 − 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.686 − 0.727i)12-s + (−0.957 − 0.286i)13-s + (−0.116 + 0.993i)14-s + (−0.802 + 0.597i)15-s + (0.766 − 0.642i)16-s + (0.642 + 0.766i)17-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.396 − 0.918i)3-s + (0.939 − 0.342i)4-s + (−0.230 − 0.973i)5-s + (−0.549 − 0.835i)6-s + (−0.286 + 0.957i)7-s + (0.866 − 0.5i)8-s + (−0.686 + 0.727i)9-s + (−0.396 − 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.686 − 0.727i)12-s + (−0.957 − 0.286i)13-s + (−0.116 + 0.993i)14-s + (−0.802 + 0.597i)15-s + (0.766 − 0.642i)16-s + (0.642 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2032091368 - 0.1596100056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2032091368 - 0.1596100056i\) |
\(L(1)\) |
\(\approx\) |
\(1.164639695 - 0.7787467115i\) |
\(L(1)\) |
\(\approx\) |
\(1.164639695 - 0.7787467115i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.396 - 0.918i)T \) |
| 5 | \( 1 + (-0.230 - 0.973i)T \) |
| 7 | \( 1 + (-0.286 + 0.957i)T \) |
| 11 | \( 1 + (0.396 - 0.918i)T \) |
| 13 | \( 1 + (-0.957 - 0.286i)T \) |
| 17 | \( 1 + (0.642 + 0.766i)T \) |
| 19 | \( 1 + (0.342 - 0.939i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (0.802 + 0.597i)T \) |
| 31 | \( 1 + (0.957 - 0.286i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.984 - 0.173i)T \) |
| 47 | \( 1 + (-0.893 + 0.448i)T \) |
| 53 | \( 1 + (-0.973 + 0.230i)T \) |
| 59 | \( 1 + (-0.802 + 0.597i)T \) |
| 61 | \( 1 + (-0.448 + 0.893i)T \) |
| 67 | \( 1 + (-0.286 - 0.957i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.396 + 0.918i)T \) |
| 79 | \( 1 + (-0.727 - 0.686i)T \) |
| 83 | \( 1 + (-0.993 - 0.116i)T \) |
| 89 | \( 1 + (-0.549 - 0.835i)T \) |
| 97 | \( 1 + (-0.727 + 0.686i)T \) |
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
−19.07779926449495152551032187506, −17.810933953863563062518637117276, −17.26998881826838332554337243727, −16.72203219901082022387405426697, −15.79349772031864986480474409119, −15.57676727528587638727754543168, −14.518955119643493051864093937, −14.30300672553586991665972594640, −13.72505299981240323328851619017, −12.46632983577833213842620615568, −11.94123486821058776634730694285, −11.44226075601871155628224044476, −10.58234865286832418980253442952, −9.86218113640541393890043759023, −9.73163512920314907245351452066, −8.04630530382804995894134594615, −7.392104280140440719106422893789, −6.81103746866863654493094999919, −6.15957897831388010123354078108, −5.24896159679279846713756677595, −4.51816035946041578273515113695, −3.94077382160937308041724687275, −3.28390374232446716733757240491, −2.60311479087979555604060221250, −1.41519575084743374973547997410,
0.02761627128580360558263680465, 0.95522702530190047233031882693, 1.64651943517352837616375556354, 2.71892806870860520409917553290, 3.10849634218704605796168218080, 4.48595999961662741107849171960, 4.95375867232123532724518991980, 5.85638071407892167879982832937, 6.13049101012156897498127266396, 7.04025149819389424766626663181, 7.97318980227814883061099987270, 8.475461081835121807103727900806, 9.415393604321640685854029496410, 10.37660623982266332870774671374, 11.287877230267225477729187531118, 11.89319864206001884276056115171, 12.38425890573825094174552971725, 12.755849979905213892317058858146, 13.51830293448508149162286476033, 14.188136327925648712730385319221, 14.93809416834557463393800278930, 15.74469431822245236963992315020, 16.39489208461160595353802400927, 16.91642099111030727157710351621, 17.680368172985499254739067953199