L(s) = 1 | + (0.183 − 0.982i)2-s + (0.273 − 0.961i)3-s + (−0.932 − 0.361i)4-s + (0.445 − 0.895i)5-s + (−0.895 − 0.445i)6-s + (−0.995 − 0.0922i)7-s + (−0.526 + 0.850i)8-s + (−0.850 − 0.526i)9-s + (−0.798 − 0.602i)10-s + (−0.602 − 0.798i)11-s + (−0.602 + 0.798i)12-s + (−0.183 − 0.982i)13-s + (−0.273 + 0.961i)14-s + (−0.739 − 0.673i)15-s + (0.739 + 0.673i)16-s + (−0.445 + 0.895i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.982i)2-s + (0.273 − 0.961i)3-s + (−0.932 − 0.361i)4-s + (0.445 − 0.895i)5-s + (−0.895 − 0.445i)6-s + (−0.995 − 0.0922i)7-s + (−0.526 + 0.850i)8-s + (−0.850 − 0.526i)9-s + (−0.798 − 0.602i)10-s + (−0.602 − 0.798i)11-s + (−0.602 + 0.798i)12-s + (−0.183 − 0.982i)13-s + (−0.273 + 0.961i)14-s + (−0.739 − 0.673i)15-s + (0.739 + 0.673i)16-s + (−0.445 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.7420366760 + 0.03822790650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.7420366760 + 0.03822790650i\) |
\(L(1)\) |
\(\approx\) |
\(0.2445292825 - 0.8163670495i\) |
\(L(1)\) |
\(\approx\) |
\(0.2445292825 - 0.8163670495i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (0.183 - 0.982i)T \) |
| 3 | \( 1 + (0.273 - 0.961i)T \) |
| 5 | \( 1 + (0.445 - 0.895i)T \) |
| 7 | \( 1 + (-0.995 - 0.0922i)T \) |
| 11 | \( 1 + (-0.602 - 0.798i)T \) |
| 13 | \( 1 + (-0.183 - 0.982i)T \) |
| 17 | \( 1 + (-0.445 + 0.895i)T \) |
| 19 | \( 1 + (-0.526 - 0.850i)T \) |
| 23 | \( 1 + (0.739 - 0.673i)T \) |
| 29 | \( 1 + (-0.0922 - 0.995i)T \) |
| 31 | \( 1 + (-0.526 + 0.850i)T \) |
| 37 | \( 1 + (-0.961 - 0.273i)T \) |
| 41 | \( 1 + (0.850 - 0.526i)T \) |
| 43 | \( 1 + (-0.995 - 0.0922i)T \) |
| 47 | \( 1 + (-0.273 + 0.961i)T \) |
| 53 | \( 1 + (0.673 + 0.739i)T \) |
| 59 | \( 1 + (-0.961 - 0.273i)T \) |
| 61 | \( 1 + (0.273 - 0.961i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.850 - 0.526i)T \) |
| 73 | \( 1 + (-0.273 - 0.961i)T \) |
| 79 | \( 1 + (0.602 - 0.798i)T \) |
| 83 | \( 1 + (0.273 - 0.961i)T \) |
| 89 | \( 1 + (0.932 - 0.361i)T \) |
| 97 | \( 1 + (0.995 + 0.0922i)T \) |
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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
−22.3234702697197672579260080731, −21.59388499812272909225748439020, −20.9865103604557878694517938625, −19.82793328036691601861262317599, −18.867591298885538473565980341074, −18.276948977982416114945891340541, −17.25017002608684824776714413869, −16.52674435825502957448645131518, −15.887290839284973074977821109020, −15.05332875879036896877063363622, −14.628372722539849087349673008808, −13.68382621609072207920481485444, −13.09994047427180691677796045782, −11.89614904656505492955073751058, −10.74244266011613970164761705138, −9.76824465912195960335572596141, −9.51918166037893979924109847427, −8.5471541842835886763165932244, −7.26817050054865093748116684737, −6.80004944791025352729455981687, −5.750725317232746935763630182197, −4.993120686276458696638248958484, −3.95483326349731870865279043061, −3.189075020611873331423432414410, −2.21710652927690406698596602224,
0.20217996821585554774836804134, 0.66272779553520012172574483314, 1.892450967767553389015333494970, 2.754756926942682352721427666973, 3.51018528929064856856435982621, 4.808965324142821957028880780931, 5.75945639198210968095153216253, 6.40146059618683874496159532720, 7.81115095518643941549892674669, 8.72118229455120821063601512685, 9.11617819545337323778128964438, 10.31345706812343241353342355431, 10.94960446019549407327079712481, 12.22286350904306169558300765655, 12.76736611841747824588815573769, 13.21704691756082358408522944612, 13.74052901547014297173949380455, 14.87735603293818980537270437802, 15.874397522513633216707584728, 17.13894570142039162198188338677, 17.515150212891952167406544030540, 18.53298809896830311349044739712, 19.27795930485006428801603415797, 19.767091828547043530159681855800, 20.47002891355413099127387650234