L(s) = 1 | + (−0.117 − 0.993i)2-s + (−0.966 + 0.257i)3-s + (−0.972 + 0.233i)4-s + (0.179 − 0.983i)5-s + (0.369 + 0.929i)6-s + (−0.944 + 0.329i)7-s + (0.346 + 0.938i)8-s + (0.867 − 0.498i)9-s + (−0.998 − 0.0620i)10-s + (−0.743 − 0.668i)11-s + (0.879 − 0.476i)12-s + (0.940 − 0.340i)13-s + (0.437 + 0.899i)14-s + (0.0806 + 0.996i)15-s + (0.890 − 0.454i)16-s + (−0.873 − 0.487i)17-s + ⋯ |
L(s) = 1 | + (−0.117 − 0.993i)2-s + (−0.966 + 0.257i)3-s + (−0.972 + 0.233i)4-s + (0.179 − 0.983i)5-s + (0.369 + 0.929i)6-s + (−0.944 + 0.329i)7-s + (0.346 + 0.938i)8-s + (0.867 − 0.498i)9-s + (−0.998 − 0.0620i)10-s + (−0.743 − 0.668i)11-s + (0.879 − 0.476i)12-s + (0.940 − 0.340i)13-s + (0.437 + 0.899i)14-s + (0.0806 + 0.996i)15-s + (0.890 − 0.454i)16-s + (−0.873 − 0.487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.002683443682 + 0.002065319707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002683443682 + 0.002065319707i\) |
\(L(1)\) |
\(\approx\) |
\(0.4055974465 - 0.2755347311i\) |
\(L(1)\) |
\(\approx\) |
\(0.4055974465 - 0.2755347311i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.117 - 0.993i)T \) |
| 3 | \( 1 + (-0.966 + 0.257i)T \) |
| 5 | \( 1 + (0.179 - 0.983i)T \) |
| 7 | \( 1 + (-0.944 + 0.329i)T \) |
| 11 | \( 1 + (-0.743 - 0.668i)T \) |
| 13 | \( 1 + (0.940 - 0.340i)T \) |
| 17 | \( 1 + (-0.873 - 0.487i)T \) |
| 19 | \( 1 + (-0.952 - 0.305i)T \) |
| 29 | \( 1 + (0.299 - 0.954i)T \) |
| 31 | \( 1 + (0.586 + 0.809i)T \) |
| 37 | \( 1 + (-0.926 + 0.375i)T \) |
| 41 | \( 1 + (-0.993 - 0.111i)T \) |
| 43 | \( 1 + (0.988 + 0.148i)T \) |
| 47 | \( 1 + (-0.334 + 0.942i)T \) |
| 53 | \( 1 + (-0.998 + 0.0620i)T \) |
| 59 | \( 1 + (0.392 - 0.919i)T \) |
| 61 | \( 1 + (-0.449 + 0.893i)T \) |
| 67 | \( 1 + (-0.0186 + 0.999i)T \) |
| 71 | \( 1 + (0.251 + 0.967i)T \) |
| 73 | \( 1 + (0.299 + 0.954i)T \) |
| 79 | \( 1 + (0.275 + 0.961i)T \) |
| 83 | \( 1 + (0.969 - 0.245i)T \) |
| 89 | \( 1 + (-0.847 - 0.530i)T \) |
| 97 | \( 1 + (-0.311 + 0.950i)T \) |
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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
−23.38184388798378407256053190658, −22.66474931732496908799072096056, −22.12537008020544925190212866213, −21.1187243060597453090400821982, −19.485662481214814779746084487405, −18.767341830749877837711491953735, −18.10908279035836678341843778132, −17.39285443285050260033630171339, −16.55158253235977047879150319739, −15.67882569293899534249323724560, −15.12743559354804140749485996050, −13.77517061326503802075843928858, −13.19781254142531763644689192555, −12.33055340918753477640103711997, −10.745995361408833465019622002756, −10.453073183013078151881417345690, −9.37948850673659876761392650590, −8.06295414048170219083725131426, −6.9789999363823039879473562374, −6.53334875409094385855406227684, −5.84385795177477162431221107371, −4.59754136410481811620708623553, −3.59538033879510186834332087911, −1.88282133929773870834731802438, −0.00247028176321163684044385737,
1.10009716323303352656396931933, 2.56742030037377180912228742954, 3.78766710209377269495097664245, 4.753785892497464533206162781573, 5.604956987634575890575107856538, 6.50086302082355479506285747455, 8.268772439059565483964633074545, 8.980043851965833231234312594830, 9.90743775546277996029132694688, 10.70185732822854891713445415125, 11.527654565748186327675824477147, 12.47811328886312552165776706575, 13.07179078507439990341809840653, 13.68172522368788911892989966550, 15.65054811741343811146771238032, 15.98838185961298268612656011915, 17.10629267889123410913718498827, 17.725664660801788850308236938599, 18.69882581941634361428491366830, 19.398545246750817042739230860067, 20.556734628925866771533030813407, 21.12514308360751143688409930300, 21.859550854582865447655094422884, 22.704311700989797974996041699111, 23.400003881042559212662931031498