| L(s) = 1 | + i·7-s − i·11-s − 13-s − i·17-s + i·23-s − i·29-s + 31-s − 37-s − 41-s + 43-s + i·47-s − 49-s − 53-s − i·59-s + i·61-s + ⋯ |
| L(s) = 1 | + i·7-s − i·11-s − 13-s − i·17-s + i·23-s − i·29-s + 31-s − 37-s − 41-s + 43-s + i·47-s − 49-s − 53-s − i·59-s + i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1508277483 + 0.4672152820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1508277483 + 0.4672152820i\) |
| \(L(1)\) |
\(\approx\) |
\(0.8624440220 + 0.07314927560i\) |
| \(L(1)\) |
\(\approx\) |
\(0.8624440220 + 0.07314927560i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−17.736150088916383350108922290729, −17.28835433038747533194808090049, −16.78728974591320801513212769491, −16.00618756059638804673745324097, −15.10809449634742594321517284488, −14.67058254846456381628354692084, −13.96704337304843919612083584599, −13.236956670158391317756642420261, −12.38718741441814347628081072384, −12.16298935262449827953459468221, −10.9784083935584430461727443314, −10.35070002579253261557524588120, −9.9948688121182754722712384881, −9.076216448860926971082576613132, −8.24440516794728025852621681080, −7.52929436644619213684809410127, −6.897553053181504594799596870045, −6.346944511935555037167560559609, −5.14579985534470957812786533864, −4.624175265496438726601097567427, −3.927361747057017908030637420286, −3.053442254125035719982798771029, −2.086830314136045105422917932, −1.37098634467694340629463425136, −0.138175888435293623311939045867,
1.06343333473982664064797306196, 2.1873641369407319354966042999, 2.807440994597126448552727023767, 3.49749747176193832875608063942, 4.63321737115322439819415525209, 5.249311479346327306548969560856, 5.91447433110212291234805463530, 6.64390813266501896210033969032, 7.57017351541020598121272519135, 8.159698749547316771731834100438, 9.0117666234045279894010749249, 9.50979230972947691860221610501, 10.230343135568869281098909631549, 11.23382175484631045394002717329, 11.75184192947347836252643494992, 12.2493038472513391341389318417, 13.14080077610406741616791959364, 13.86337308312263827239174666087, 14.37479655422699794178152881059, 15.322975722582931897735984848576, 15.73088362578946177558799990035, 16.374567617532090896108955078493, 17.3745098709662584350151963942, 17.62733774688372792943524622092, 18.81293791425265950074111557279
