L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + i·6-s + (0.5 + 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + i·6-s + (0.5 + 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2111509126 - 0.5602309438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2111509126 - 0.5602309438i\) |
\(L(1)\) |
\(\approx\) |
\(0.5192689671 - 0.2410718148i\) |
\(L(1)\) |
\(\approx\) |
\(0.5192689671 - 0.2410718148i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 - iT \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.866 + 0.5i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.866 - 0.5i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 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
−27.284742228134925759736321647462, −26.59350598300178296262550500290, −25.75363494726249033646635571630, −24.52653567795884446946248401052, −23.49544319986717950793839680920, −22.89613055172104587055365693414, −21.392732095024155166734113963092, −20.534116702028319043337572019468, −19.73655037429677292925961670682, −18.22330956017463065862471659943, −17.591583109130148375235077467172, −16.60023567615190693283936070091, −15.92452448594475365048093413748, −14.85524446041454664939523980101, −13.97793575802533620959370788117, −12.135151687098587904891119847155, −10.95070497238277332376429457150, −10.23961179597457592306339291177, −9.45386495682402592841433631683, −7.96637461590062719405766610470, −7.19406058628921000689152337316, −5.55193552370089649007297123199, −4.92950342771700650721872287148, −3.13255396471413268729603358383, −1.07640298806854312236130748427,
0.35036964689769778621081409131, 1.8990531021933293129159102872, 2.76416154437764369466726816411, 4.91656255932835176610908733090, 6.1981790552583496866443453332, 7.56171419272647666021920467682, 8.15685101314802800816177874338, 9.500866742343337103654526327684, 10.69885985589518460301519538604, 11.75163110380171473560087405715, 12.3317022512837839053006572675, 13.41002539261098354902843244505, 14.93288127103248726345735736723, 16.243300309605454283235643619430, 17.15749289692987893112313064099, 18.12739315500535262763922258456, 18.69259362484087845704583108777, 19.52748585770733205403195208577, 20.791724487963237110288386099490, 21.67680454567230011182986480693, 22.667768622817933468655766851814, 24.122441689672360352471056311541, 24.62868246379508934530526146351, 25.74346180266276378386607969431, 26.65728119848216061882998748267