L(s) = 1 | + (−0.886 + 0.462i)2-s + (0.0922 + 0.995i)3-s + (0.572 − 0.819i)4-s + (−0.542 − 0.840i)5-s + (−0.542 − 0.840i)6-s + (0.997 + 0.0738i)7-s + (−0.128 + 0.991i)8-s + (−0.982 + 0.183i)9-s + (0.869 + 0.494i)10-s + (−0.412 + 0.911i)11-s + (0.869 + 0.494i)12-s + (0.445 + 0.895i)13-s + (−0.918 + 0.395i)14-s + (0.786 − 0.617i)15-s + (−0.343 − 0.938i)16-s + (−0.0554 + 0.998i)17-s + ⋯ |
L(s) = 1 | + (−0.886 + 0.462i)2-s + (0.0922 + 0.995i)3-s + (0.572 − 0.819i)4-s + (−0.542 − 0.840i)5-s + (−0.542 − 0.840i)6-s + (0.997 + 0.0738i)7-s + (−0.128 + 0.991i)8-s + (−0.982 + 0.183i)9-s + (0.869 + 0.494i)10-s + (−0.412 + 0.911i)11-s + (0.869 + 0.494i)12-s + (0.445 + 0.895i)13-s + (−0.918 + 0.395i)14-s + (0.786 − 0.617i)15-s + (−0.343 − 0.938i)16-s + (−0.0554 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.005038020521 + 0.6043648133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.005038020521 + 0.6043648133i\) |
\(L(1)\) |
\(\approx\) |
\(0.5337618851 + 0.3640108928i\) |
\(L(1)\) |
\(\approx\) |
\(0.5337618851 + 0.3640108928i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.886 + 0.462i)T \) |
| 3 | \( 1 + (0.0922 + 0.995i)T \) |
| 5 | \( 1 + (-0.542 - 0.840i)T \) |
| 7 | \( 1 + (0.997 + 0.0738i)T \) |
| 11 | \( 1 + (-0.412 + 0.911i)T \) |
| 13 | \( 1 + (0.445 + 0.895i)T \) |
| 17 | \( 1 + (-0.0554 + 0.998i)T \) |
| 19 | \( 1 + (-0.982 - 0.183i)T \) |
| 23 | \( 1 + (-0.343 + 0.938i)T \) |
| 29 | \( 1 + (0.997 - 0.0738i)T \) |
| 31 | \( 1 + (0.687 - 0.726i)T \) |
| 37 | \( 1 + (0.975 + 0.219i)T \) |
| 41 | \( 1 + (-0.982 - 0.183i)T \) |
| 43 | \( 1 + (0.378 - 0.925i)T \) |
| 47 | \( 1 + (-0.918 + 0.395i)T \) |
| 53 | \( 1 + (0.830 - 0.557i)T \) |
| 59 | \( 1 + (-0.659 - 0.751i)T \) |
| 61 | \( 1 + (-0.918 + 0.395i)T \) |
| 67 | \( 1 + (0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.128 + 0.991i)T \) |
| 73 | \( 1 + (-0.918 - 0.395i)T \) |
| 79 | \( 1 + (-0.993 + 0.110i)T \) |
| 83 | \( 1 + (0.0922 + 0.995i)T \) |
| 89 | \( 1 + (-0.602 + 0.798i)T \) |
| 97 | \( 1 + (-0.850 + 0.526i)T \) |
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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
−21.02418305755211226987175633488, −20.16180001358350005257492209649, −19.56108050300249889640114226226, −18.66766336294841762392554076777, −18.23080527461149679221974026146, −17.77434950426493500007471917914, −16.73637173369772460783327308248, −15.83612819055858626511228494915, −14.8827313207001392493045819727, −14.03103767546422912034411015045, −13.18245039211726193288531401375, −12.17373819114137433795984883557, −11.5308984153293867028739441769, −10.852516927027303343491276916363, −10.272351555088852267475908952960, −8.63256374593006167404698596580, −8.26891862844195727862679014149, −7.623793556172104447678489774346, −6.73481841951147184064542048682, −5.93727340707463158958729516154, −4.40711148659089679613772839549, −2.99799382718305794231844813685, −2.688964558500757133796103362099, −1.398128925021239316448745424149, −0.35587364314230254782583443322,
1.41312937898299695859143051498, 2.306880192632144184288940078855, 4.03265457245132318264971095700, 4.62647715855825308663413430908, 5.43144111368212336877606404333, 6.485201773551090937511902260, 7.80415853189504442114445006441, 8.30072164107563709751821468375, 8.96765484747949852055396871658, 9.813426980354314276314958454526, 10.64758366885030279684809591194, 11.43036235235824020349428335483, 12.078047711362255394297842051170, 13.485938692364708788518321769692, 14.51873901727221406161323811752, 15.29846203091736237498737008631, 15.583490423283085976943045499949, 16.62434387704525476421180245196, 17.17692952724176859076829498026, 17.78444708993464398982310810345, 18.94793996812749640873660846715, 19.72231404217938454345633345050, 20.380725892811922248147490816879, 21.09203339087600794239010260199, 21.58295440675650907131334078428