L(s) = 1 | + (−0.968 − 0.248i)3-s + (0.929 − 0.368i)7-s + (0.876 + 0.481i)9-s + (0.876 + 0.481i)11-s + (0.929 + 0.368i)13-s + (−0.309 + 0.951i)17-s + (0.637 − 0.770i)19-s + (−0.992 + 0.125i)21-s + (0.0627 − 0.998i)23-s + (−0.728 − 0.684i)27-s + (0.929 + 0.368i)29-s + (0.929 − 0.368i)31-s + (−0.728 − 0.684i)33-s + (−0.968 + 0.248i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
L(s) = 1 | + (−0.968 − 0.248i)3-s + (0.929 − 0.368i)7-s + (0.876 + 0.481i)9-s + (0.876 + 0.481i)11-s + (0.929 + 0.368i)13-s + (−0.309 + 0.951i)17-s + (0.637 − 0.770i)19-s + (−0.992 + 0.125i)21-s + (0.0627 − 0.998i)23-s + (−0.728 − 0.684i)27-s + (0.929 + 0.368i)29-s + (0.929 − 0.368i)31-s + (−0.728 − 0.684i)33-s + (−0.968 + 0.248i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.469250331 - 1.293485317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469250331 - 1.293485317i\) |
\(L(1)\) |
\(\approx\) |
\(1.008713438 - 0.1842851528i\) |
\(L(1)\) |
\(\approx\) |
\(1.008713438 - 0.1842851528i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 101 | \( 1 \) |
good | 3 | \( 1 + (-0.968 - 0.248i)T \) |
| 7 | \( 1 + (0.929 - 0.368i)T \) |
| 11 | \( 1 + (0.876 + 0.481i)T \) |
| 13 | \( 1 + (0.929 + 0.368i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.637 - 0.770i)T \) |
| 23 | \( 1 + (0.0627 - 0.998i)T \) |
| 29 | \( 1 + (0.929 + 0.368i)T \) |
| 31 | \( 1 + (0.929 - 0.368i)T \) |
| 37 | \( 1 + (-0.968 + 0.248i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 + (-0.187 - 0.982i)T \) |
| 47 | \( 1 + (-0.187 + 0.982i)T \) |
| 53 | \( 1 + (-0.425 - 0.904i)T \) |
| 59 | \( 1 + (-0.637 - 0.770i)T \) |
| 61 | \( 1 + (0.425 - 0.904i)T \) |
| 67 | \( 1 + (-0.968 + 0.248i)T \) |
| 71 | \( 1 + (-0.968 - 0.248i)T \) |
| 73 | \( 1 + (0.0627 - 0.998i)T \) |
| 79 | \( 1 + (-0.0627 - 0.998i)T \) |
| 83 | \( 1 + (-0.0627 - 0.998i)T \) |
| 89 | \( 1 + (0.637 - 0.770i)T \) |
| 97 | \( 1 + (0.425 - 0.904i)T \) |
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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
−19.98936335243001643130623523119, −19.024291126531318389501339882455, −18.17785594627729866840882649868, −17.83755774944400932016772228971, −17.0750777107010567646693909767, −16.23811000557573474352636551404, −15.701238800363272178231444153005, −14.94457131685700344141579713836, −13.96114777565952314003558280448, −13.44432127260287466524760667397, −12.15894141435954478321546561460, −11.75040306268635947178371854026, −11.21561158750704029124971311419, −10.392930862196727279724528405, −9.551117986510177577675180156174, −8.7206407798204348623108321550, −7.91784006473342445267940247824, −6.938659704452523597852449085938, −6.1151173401360174093277221105, −5.475278481949313320687484062097, −4.71288744637139940551810277143, −3.876576450886178754518509789841, −2.92673242483172521847426369322, −1.394269093121833301773294586605, −1.05263921578199612566165547544,
0.4534682781414826655284919666, 1.380417747625360760509259157317, 1.96397583083720185613135871288, 3.51218864112370063662987123315, 4.50449584504473542920057648380, 4.848308725053782320452376594487, 6.06589687205568727326069882008, 6.620128374048519217823596747680, 7.3408148052580178904204165798, 8.33695034772876005417581748299, 9.01185731334766095846391690162, 10.24726355648578369355959410199, 10.70680768811243473243561169491, 11.552785466125881594203926131239, 11.99206932720180685037184368801, 12.88302836076245543212862128736, 13.734486751718295157679655991977, 14.344126575200268606658194243688, 15.349436828391977927780988091585, 15.98226733096514698773252657729, 16.914755236735116049320837765438, 17.468795729014839364948717127134, 17.86358193361499258505874072567, 18.7956575273361384888547948857, 19.43586492781396942503489804716