L(s) = 1 | + (−0.946 + 0.321i)3-s + (−0.896 + 0.442i)5-s + (0.793 − 0.608i)9-s + (0.659 + 0.751i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.258 + 0.965i)17-s + (0.997 − 0.0654i)19-s + (0.608 + 0.793i)23-s + (0.608 − 0.793i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (−0.442 − 0.896i)37-s + ⋯ |
L(s) = 1 | + (−0.946 + 0.321i)3-s + (−0.896 + 0.442i)5-s + (0.793 − 0.608i)9-s + (0.659 + 0.751i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.258 + 0.965i)17-s + (0.997 − 0.0654i)19-s + (0.608 + 0.793i)23-s + (0.608 − 0.793i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (−0.442 − 0.896i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7483640939 + 0.8693837301i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7483640939 + 0.8693837301i\) |
\(L(1)\) |
\(\approx\) |
\(0.7028797049 + 0.2332458127i\) |
\(L(1)\) |
\(\approx\) |
\(0.7028797049 + 0.2332458127i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.946 + 0.321i)T \) |
| 5 | \( 1 + (-0.896 + 0.442i)T \) |
| 11 | \( 1 + (0.659 + 0.751i)T \) |
| 13 | \( 1 + (-0.831 + 0.555i)T \) |
| 17 | \( 1 + (-0.258 + 0.965i)T \) |
| 19 | \( 1 + (0.997 - 0.0654i)T \) |
| 23 | \( 1 + (0.608 + 0.793i)T \) |
| 29 | \( 1 + (-0.980 - 0.195i)T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (-0.442 - 0.896i)T \) |
| 41 | \( 1 + (0.382 - 0.923i)T \) |
| 43 | \( 1 + (0.195 + 0.980i)T \) |
| 47 | \( 1 + (0.965 - 0.258i)T \) |
| 53 | \( 1 + (0.659 + 0.751i)T \) |
| 59 | \( 1 + (0.0654 - 0.997i)T \) |
| 61 | \( 1 + (0.751 + 0.659i)T \) |
| 67 | \( 1 + (0.946 - 0.321i)T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 + (-0.130 - 0.991i)T \) |
| 79 | \( 1 + (-0.258 - 0.965i)T \) |
| 83 | \( 1 + (0.555 + 0.831i)T \) |
| 89 | \( 1 + (0.991 + 0.130i)T \) |
| 97 | \( 1 - iT \) |
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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
−21.79195256841591694647464938085, −20.621484186098410928301612382971, −19.941652187438542557819130139304, −19.02981140660986729939530572389, −18.48712661052283669178951977355, −17.412046876691648913340467461644, −16.77724362308600725739532148916, −16.09536336377554145936245325315, −15.38809421881028034221884359193, −14.28730767563912256579847153467, −13.30449542781668834052708426695, −12.462639732596422324567170414757, −11.75397368628612877756883082618, −11.28668071698071092125706021743, −10.25613031078639272762876878297, −9.22644781104303856652281732255, −8.229982152144445024624266332396, −7.32309694966935446985740200524, −6.69845119651422581491554413541, −5.42850410294447792237748634927, −4.89926798661053818224309460807, −3.836260256577965923005087192770, −2.696137533576152171473029347681, −1.07283142256145918086383635136, −0.48486215661724895939478797458,
0.77056102459072608941289189815, 2.08219691694861395131345551627, 3.588789421754587266656016275053, 4.21903017561992545798821771664, 5.110561625020084892342467026296, 6.1842557472218070478048835583, 7.11946960345014693369867319555, 7.58159661279627849832845453723, 9.07448126066696213818998826385, 9.78843348721731965967211709502, 10.7433826806532418441840335928, 11.50538893678572273886877637152, 12.067422661104508384987478550088, 12.797954709392327859516421572247, 14.12675851812982650768647762704, 15.07086318084814562304378718677, 15.4761354121717795171124189116, 16.46955034548171046095958085976, 17.226739945551181279032019495062, 17.81536202956265584773209063277, 18.903700249099905595895946420963, 19.47153759795786637178582477381, 20.375516947755532057286884858890, 21.37015950587974838499622533723, 22.22305034196412698096043850363