L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·13-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)23-s + i·27-s + 29-s + (0.5 + 0.866i)31-s + (0.866 + 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + 41-s − i·43-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·13-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)23-s + i·27-s + 29-s + (0.5 + 0.866i)31-s + (0.866 + 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + 41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9056956267 + 0.01512635008i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9056956267 + 0.01512635008i\) |
\(L(1)\) |
\(\approx\) |
\(0.8421968431 + 0.04511952603i\) |
\(L(1)\) |
\(\approx\) |
\(0.8421968431 + 0.04511952603i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 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
−25.37916858690717815669916241289, −24.771896595574849001135429210168, −23.56565809898251496051028671274, −23.0713457496231216503449613296, −22.21070685028351255514029549834, −21.16298606172786333046880531830, −20.13793111461551787575495023227, −19.102502477305461489370669435627, −18.081991520338193445942768912573, −17.58692283047981425045744485266, −16.4862892498665274622577476753, −15.62971205161765521790827899510, −14.49196846526787872622968052688, −13.253213041321129619827655361935, −12.47242638224329929152987304190, −11.72268911433610104396918196595, −10.430525810109869636837641745812, −9.88260430723860392807905028181, −8.01601179014521219994223269845, −7.497285820051615223988839534491, −6.08033727989539212553514472336, −5.39814211242563379859592432392, −4.12336686886065819204820742573, −2.49587398027428853189327406215, −1.10398270488693037058004677493,
0.88869719583456950788523523570, 2.80448570502580121670879938084, 4.13492534224405015441688736722, 5.15213719568540117388191531781, 6.126620653925983041245902901691, 7.15982848555647717450619192928, 8.55889776133390155440468662493, 9.65170084422078113181500951934, 10.56246974623367299712337846115, 11.53957778314267913182334279650, 12.22332026461603908869947700577, 13.57272715140847985316974109961, 14.47255124113908834119280011158, 15.90894253134805311282731050972, 16.19860609285451312282518165919, 17.2897777994744258999924396105, 18.2421324061940764498113298514, 19.04807061284631746268086073270, 20.33842653144171260057897868738, 21.38552194537092507803626584128, 21.82539318259992666132333784962, 22.948132366537596994898805853999, 23.74377230482364539657856359513, 24.42941976117095080598383761217, 25.81546672136811417033014990014