L(s) = 1 | + (0.984 + 0.173i)5-s + (0.984 − 0.173i)11-s + (−0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.984 + 0.173i)29-s + (−0.173 + 0.984i)31-s − i·37-s + (0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)5-s + (0.984 − 0.173i)11-s + (−0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.984 + 0.173i)29-s + (−0.173 + 0.984i)31-s − i·37-s + (0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.998517149 - 0.4774830227i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.998517149 - 0.4774830227i\) |
\(L(1)\) |
\(\approx\) |
\(1.355616583 + 0.004875017364i\) |
\(L(1)\) |
\(\approx\) |
\(1.355616583 + 0.004875017364i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.984 + 0.173i)T \) |
| 11 | \( 1 + (0.984 - 0.173i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.984 + 0.173i)T \) |
| 31 | \( 1 + (-0.173 + 0.984i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.984 - 0.173i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.984 - 0.173i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.766 + 0.642i)T \) |
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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
−18.83329477628542171973583450478, −18.08090292288847975491204091453, −17.56921676991607692041207203365, −16.70144193814166907353428085931, −16.48274891444828455239312188196, −15.269714517061359030618618683199, −14.64131205619805743519475436359, −13.95261843642455789405574344223, −13.4274950064367621788471976293, −12.58830578668135216384022794826, −11.7384475655607658547912462319, −11.353404171312994706385830448317, −10.04644755677516606887727851559, −9.55308346318751000674907317690, −9.292566423879624915636275578525, −8.07600070237453834759006485922, −7.30616372555321270366862611352, −6.59181932760790335732488249532, −5.7618886510891565355190402187, −5.08569998965442813277305344007, −4.32729070704629212162124092647, −3.25486756537638330918917471915, −2.42645775204080859276083829887, −1.59017395576085944638457682388, −0.76151791989979561924444811604,
0.57699452494771604769162917736, 1.666995407601423940654197483522, 2.19542175455314027369002167688, 3.354225613593665716045450120526, 3.96632882633134382995523312687, 5.22142827930784868745116099055, 5.62284829194757945791572990222, 6.53365599212561268897492747313, 7.15946741334274371114218081288, 8.07631899661911187832352693554, 8.984755362717030760898512910, 9.57395345929896674319955909120, 10.28472095935218907809091053188, 10.83784292493618512893362769794, 12.05760504947412404630393776575, 12.31455045663362666304320184058, 13.28595277630194894242180631009, 14.11750280472415032794777050617, 14.47756598161342149740483262392, 15.13429735230316611565349919179, 16.31559774646355207280957504986, 16.78223794871470540161077460791, 17.43391059265141196937377577591, 18.0698905013619070994938237714, 18.79650582741801613975559954025