L(s) = 1 | + (0.587 − 0.809i)3-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.951 + 0.309i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.951 + 0.309i)37-s + (−0.309 + 0.951i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.951 + 0.309i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.951 + 0.309i)37-s + (−0.309 + 0.951i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4512950510 - 1.139841185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4512950510 - 1.139841185i\) |
\(L(1)\) |
\(\approx\) |
\(0.9941848849 - 0.4309619501i\) |
\(L(1)\) |
\(\approx\) |
\(0.9941848849 - 0.4309619501i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (-0.587 - 0.809i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.951 + 0.309i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.587 - 0.809i)T \) |
| 53 | \( 1 + (-0.587 + 0.809i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.587 - 0.809i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−20.99987166681654402979238587844, −20.484147404487459192992119477698, −19.43647825840798680324887660349, −19.206963463236416191073011343, −18.05339974016026334342767830108, −17.1557541521867351606323567051, −16.418480170314906187264984019066, −15.81167544082619940077836547685, −14.89762557438842907645185791004, −14.44563089033201496749181592767, −13.510510623429541818745276449637, −12.841297815378406294042559286103, −11.75120339616594842840692048665, −10.84500487068708928903862877430, −10.29903910939586715076145417295, −9.41882514690118605164617009630, −8.65043430007515975523890519139, −7.980469179491003368987067593786, −7.07437813440910907321916033591, −5.84805587467913535511991111196, −5.11459728097848156448372767128, −4.26698307979619721263234887434, −3.22766255568610635783824879575, −2.71299168936603289248923759786, −1.4056968604369863824533156970,
0.4127344794852048719308669054, 1.80981660654911136900887146237, 2.46379720688841748994860641602, 3.35685153472115292461044173130, 4.57937859511696867768333662327, 5.32629196402759861906978156617, 6.5976260055080311662440058047, 7.28172730798575533138469641815, 7.6820839487633080613757488614, 8.97824237005579637347618723055, 9.38235561925317433594364196884, 10.3341290152516663813027131123, 11.594968851853782982918721522848, 12.04316093080944915791872850775, 12.964417187739363544296335165442, 13.61079698767344899983241447825, 14.28306956951906216659046472055, 15.29946708217982911457227369691, 15.58424057549010180364064376309, 17.17365621474922722268016963276, 17.36734125047881231743386998405, 18.43135223593970722363423046492, 18.93357079435252645135477568828, 19.79895313739353956035406104556, 20.42546932171634907922494162104