L(s) = 1 | + (0.895 + 0.445i)2-s + (−0.671 − 0.740i)3-s + (0.603 + 0.797i)4-s + (−0.589 − 0.807i)5-s + (−0.271 − 0.962i)6-s + (0.421 − 0.906i)7-s + (0.185 + 0.982i)8-s + (−0.0974 + 0.995i)9-s + (−0.167 − 0.985i)10-s + (0.254 + 0.967i)11-s + (0.185 − 0.982i)12-s + (0.574 − 0.818i)13-s + (0.781 − 0.624i)14-s + (−0.202 + 0.979i)15-s + (−0.271 + 0.962i)16-s + (0.710 + 0.703i)17-s + ⋯ |
L(s) = 1 | + (0.895 + 0.445i)2-s + (−0.671 − 0.740i)3-s + (0.603 + 0.797i)4-s + (−0.589 − 0.807i)5-s + (−0.271 − 0.962i)6-s + (0.421 − 0.906i)7-s + (0.185 + 0.982i)8-s + (−0.0974 + 0.995i)9-s + (−0.167 − 0.985i)10-s + (0.254 + 0.967i)11-s + (0.185 − 0.982i)12-s + (0.574 − 0.818i)13-s + (0.781 − 0.624i)14-s + (−0.202 + 0.979i)15-s + (−0.271 + 0.962i)16-s + (0.710 + 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.834072381 - 0.6251476728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834072381 - 0.6251476728i\) |
\(L(1)\) |
\(\approx\) |
\(1.440832986 - 0.1749951011i\) |
\(L(1)\) |
\(\approx\) |
\(1.440832986 - 0.1749951011i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 709 | \( 1 \) |
good | 2 | \( 1 + (0.895 + 0.445i)T \) |
| 3 | \( 1 + (-0.671 - 0.740i)T \) |
| 5 | \( 1 + (-0.589 - 0.807i)T \) |
| 7 | \( 1 + (0.421 - 0.906i)T \) |
| 11 | \( 1 + (0.254 + 0.967i)T \) |
| 13 | \( 1 + (0.574 - 0.818i)T \) |
| 17 | \( 1 + (0.710 + 0.703i)T \) |
| 19 | \( 1 + (0.758 - 0.651i)T \) |
| 23 | \( 1 + (-0.981 - 0.194i)T \) |
| 29 | \( 1 + (0.999 - 0.0354i)T \) |
| 31 | \( 1 + (-0.746 - 0.665i)T \) |
| 37 | \( 1 + (0.421 - 0.906i)T \) |
| 41 | \( 1 + (-0.560 - 0.828i)T \) |
| 43 | \( 1 + (0.545 + 0.838i)T \) |
| 47 | \( 1 + (0.658 - 0.752i)T \) |
| 53 | \( 1 + (0.949 - 0.314i)T \) |
| 59 | \( 1 + (0.185 + 0.982i)T \) |
| 61 | \( 1 + (0.00887 - 0.999i)T \) |
| 67 | \( 1 + (0.0443 - 0.999i)T \) |
| 71 | \( 1 + (-0.167 + 0.985i)T \) |
| 73 | \( 1 + (-0.132 - 0.991i)T \) |
| 79 | \( 1 + (0.355 - 0.934i)T \) |
| 83 | \( 1 + (-0.339 - 0.940i)T \) |
| 89 | \( 1 + (-0.852 + 0.522i)T \) |
| 97 | \( 1 + (-0.973 - 0.228i)T \) |
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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
−22.36259053619796722675708502904, −22.04342567348478006412704721026, −21.30974111661207020763294162983, −20.60608754922383533123517265239, −19.51731169237176186470305134905, −18.54055914386760192232914881094, −18.21727253877901663824744686087, −16.52215557117722616600132498052, −15.97557299568273722280888834744, −15.334475010418839551861792757861, −14.27333300949800373539134308460, −14.016754985372912008963497431568, −12.30068700143102366497687841912, −11.70411170637287802279197462451, −11.35681741609857444811897250210, −10.41370766106773128468238247627, −9.54976963193328891751723486727, −8.37864001415743538320429446679, −6.99272451397476609173291965064, −6.043028800800954438795460460, −5.49294829900613869100466121812, −4.363342362244950355579480988745, −3.53667831653535161379194550283, −2.78834993549952316272991914231, −1.236795445476925173355729531226,
0.89307778170403720159778778013, 2.00929383089478633092580143873, 3.62776527465967329780203713207, 4.44216496882973438226947853613, 5.25905290634975378379438468106, 6.10402487302121280575256886738, 7.32277756266821396448696733551, 7.652254131295098228362117207688, 8.54194236978458246415938997971, 10.238597374877046570509425904506, 11.1778691478410789767895408363, 11.99732202421258697174249436744, 12.59967028440786138693169629294, 13.31114196161131217030387795557, 14.10489109041603628079603614499, 15.14843571432187430711809058495, 16.060703810253098810249540196463, 16.72892790755095683475022817935, 17.48886870528718906982442945264, 18.07150993313499290617411272975, 19.66476455557182736430275785135, 20.10797199023120866457296182677, 20.87204293461250165855936600856, 21.984028357690654276233833886263, 22.92562124467900913341771994240