L(s) = 1 | + (0.955 + 0.294i)2-s + (0.826 + 0.563i)4-s + (−0.623 − 0.781i)5-s + (0.623 + 0.781i)8-s + (−0.365 − 0.930i)10-s + (−0.222 − 0.974i)11-s + (−0.955 − 0.294i)13-s + (0.365 + 0.930i)16-s + (−0.826 + 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.826 − 0.563i)26-s + ⋯ |
L(s) = 1 | + (0.955 + 0.294i)2-s + (0.826 + 0.563i)4-s + (−0.623 − 0.781i)5-s + (0.623 + 0.781i)8-s + (−0.365 − 0.930i)10-s + (−0.222 − 0.974i)11-s + (−0.955 − 0.294i)13-s + (0.365 + 0.930i)16-s + (−0.826 + 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.826 − 0.563i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3711291685 + 1.195607175i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3711291685 + 1.195607175i\) |
\(L(1)\) |
\(\approx\) |
\(1.317997887 + 0.2864175172i\) |
\(L(1)\) |
\(\approx\) |
\(1.317997887 + 0.2864175172i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.955 + 0.294i)T \) |
| 5 | \( 1 + (-0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.955 - 0.294i)T \) |
| 17 | \( 1 + (-0.826 + 0.563i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 29 | \( 1 + (0.0747 + 0.997i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.0747 + 0.997i)T \) |
| 41 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (-0.988 - 0.149i)T \) |
| 47 | \( 1 + (-0.955 - 0.294i)T \) |
| 53 | \( 1 + (0.0747 - 0.997i)T \) |
| 59 | \( 1 + (0.988 + 0.149i)T \) |
| 61 | \( 1 + (-0.826 + 0.563i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.733 + 0.680i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.955 + 0.294i)T \) |
| 89 | \( 1 + (-0.955 + 0.294i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−23.30809054525295485117322284824, −22.528825265615462266887210308223, −22.10366018127763369504325503792, −21.012231165443841911870797400966, −19.97214131451415928681865279542, −19.58210672237601060313862502261, −18.46002157167710005629900332612, −17.545488981222786453705699909388, −16.15236122729129618282036356687, −15.421125778759881354122488979995, −14.75156881480124080931763253675, −13.91510668359441356657515153524, −12.91828742719754106696008782904, −11.90382638431445826069597195348, −11.3954958430316965784383578275, −10.29761317105969322914622249617, −9.50678657769651485046055202530, −7.71183852167352858025570291481, −7.089909863998218615804690825156, −6.16167053995466495262859483418, −4.740838959206092400471300479132, −4.20070043995485947755170593933, −2.79453013368848886322387438518, −2.175333066166960540777927868753, −0.21618682090755937800637069308,
1.514073278296798712096893798470, 2.993971233269836887403597390812, 3.91806465302329723564534366596, 4.93762288624505453573298623715, 5.70597872105962057516516439329, 6.88490725090194246324174172472, 7.980638340040671641350988653692, 8.54965234573103437613898556530, 10.08299332738420515486721838448, 11.251176040277237179027042127, 12.033438085264696783128723272282, 12.77565590208799414395927491156, 13.63754108337481526478081727551, 14.57723091849285933191879820405, 15.521111561583607771279129201262, 16.24379430558017981988758804256, 16.890972914678337032844339076889, 17.99439443328327467946685069740, 19.491629060615748918474990810284, 19.91125924875646382958014946829, 20.92314843103966306310536645997, 21.72377021052342859880044882152, 22.493268349744837758079542723289, 23.47409829709664558469990448368, 24.25199897483882770026485953655