L(s) = 1 | + (0.984 + 0.173i)2-s + (−0.984 + 0.173i)3-s + (0.939 + 0.342i)4-s − 6-s + (−0.642 − 0.766i)7-s + (0.866 + 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.984 − 0.173i)12-s + (−0.342 + 0.939i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)18-s + (0.173 + 0.984i)19-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)2-s + (−0.984 + 0.173i)3-s + (0.939 + 0.342i)4-s − 6-s + (−0.642 − 0.766i)7-s + (0.866 + 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.984 − 0.173i)12-s + (−0.342 + 0.939i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)18-s + (0.173 + 0.984i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2909040746 + 1.086407662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2909040746 + 1.086407662i\) |
\(L(1)\) |
\(\approx\) |
\(1.057760853 + 0.3684891841i\) |
\(L(1)\) |
\(\approx\) |
\(1.057760853 + 0.3684891841i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 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
−26.53177894787246926282779544358, −25.29086785295084965439212928241, −24.30008489889106368460548405795, −23.74373754191063350293354999951, −22.48621520387662136247271880505, −22.089864589532331370042726718915, −21.25445355333136371013171069176, −19.85128321564882256435065774359, −18.94710465320323777483831015225, −17.85345915218275227037466782460, −16.54179350628714314138038704431, −15.77444577886358275879217094026, −14.92837434526177186817089335712, −13.25602505392624050043435347855, −12.84451660626450989827491538097, −11.753268310576973465222633375089, −10.87754848401857622863661633788, −9.88253418440121408647141881353, −8.03317798675461602997612601182, −6.59431328214339881187579627893, −5.83070503588483482626604459254, −4.98947764374517235985723583336, −3.483035074403805509065681785731, −2.16716377028459915771239964517, −0.29709161892293895924452696448,
1.81040952614667380593203180056, 3.62948607506872931453068111410, 4.58910025714515911987897933049, 5.614754019931103049805596526056, 6.85307840796494004884082202469, 7.40866373832858619957456392927, 9.653794794849273939372221943755, 10.57076600432056388405121962414, 11.69932477525400839265366145414, 12.501407411281436062342359007777, 13.46787846874616620394068949567, 14.56901769527110556811076171037, 15.873168572290311973359026022125, 16.398914826567522041318018899692, 17.344630479895055529088734868041, 18.58203108463388060394159463595, 20.04825921232351809697614097144, 20.81165682295479646739499476659, 22.00132148673689014777136080944, 22.61330100262249430051034239255, 23.500344405365748915249904719, 24.03909995185673905529350122382, 25.38033108869489386624903960886, 26.29486939155764823678644093124, 27.38205866794001637921076210395