L(s) = 1 | + (−0.875 − 0.483i)2-s + (−0.994 + 0.104i)3-s + (0.532 + 0.846i)4-s + (0.921 + 0.389i)6-s + (−0.0570 − 0.998i)8-s + (0.978 − 0.207i)9-s + (−0.618 − 0.786i)12-s + (−0.170 + 0.985i)13-s + (−0.432 + 0.901i)16-s + (−0.803 − 0.595i)17-s + (−0.956 − 0.290i)18-s + (0.00951 + 0.999i)19-s + (−0.690 + 0.723i)23-s + (0.161 + 0.986i)24-s + (0.625 − 0.780i)26-s + (−0.951 + 0.309i)27-s + ⋯ |
L(s) = 1 | + (−0.875 − 0.483i)2-s + (−0.994 + 0.104i)3-s + (0.532 + 0.846i)4-s + (0.921 + 0.389i)6-s + (−0.0570 − 0.998i)8-s + (0.978 − 0.207i)9-s + (−0.618 − 0.786i)12-s + (−0.170 + 0.985i)13-s + (−0.432 + 0.901i)16-s + (−0.803 − 0.595i)17-s + (−0.956 − 0.290i)18-s + (0.00951 + 0.999i)19-s + (−0.690 + 0.723i)23-s + (0.161 + 0.986i)24-s + (0.625 − 0.780i)26-s + (−0.951 + 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.007569714972 + 0.1962525860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.007569714972 + 0.1962525860i\) |
\(L(1)\) |
\(\approx\) |
\(0.4595108051 + 0.02704980691i\) |
\(L(1)\) |
\(\approx\) |
\(0.4595108051 + 0.02704980691i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.875 - 0.483i)T \) |
| 3 | \( 1 + (-0.994 + 0.104i)T \) |
| 13 | \( 1 + (-0.170 + 0.985i)T \) |
| 17 | \( 1 + (-0.803 - 0.595i)T \) |
| 19 | \( 1 + (0.00951 + 0.999i)T \) |
| 23 | \( 1 + (-0.690 + 0.723i)T \) |
| 29 | \( 1 + (-0.198 + 0.980i)T \) |
| 31 | \( 1 + (0.830 - 0.556i)T \) |
| 37 | \( 1 + (-0.244 + 0.969i)T \) |
| 41 | \( 1 + (-0.0855 + 0.996i)T \) |
| 43 | \( 1 + (-0.540 + 0.841i)T \) |
| 47 | \( 1 + (0.956 - 0.290i)T \) |
| 53 | \( 1 + (0.901 - 0.432i)T \) |
| 59 | \( 1 + (-0.905 - 0.424i)T \) |
| 61 | \( 1 + (0.999 + 0.0190i)T \) |
| 67 | \( 1 + (-0.945 + 0.327i)T \) |
| 71 | \( 1 + (-0.736 + 0.676i)T \) |
| 73 | \( 1 + (0.647 + 0.761i)T \) |
| 79 | \( 1 + (0.710 - 0.703i)T \) |
| 83 | \( 1 + (-0.999 + 0.0285i)T \) |
| 89 | \( 1 + (-0.995 + 0.0950i)T \) |
| 97 | \( 1 + (0.633 + 0.774i)T \) |
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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
−17.97514274403489884887468699282, −17.27854538502938739812974580762, −17.01096293291599170199819956793, −16.03069775499050284402699359757, −15.48141343894008399403625573472, −15.134611018212019547829853863475, −13.96972794149472989426121018088, −13.301739822498551009402067780349, −12.36537916449373529875114813633, −11.83626052443972496185365472913, −10.845960584472217367568448209852, −10.592593703198297892539955184353, −9.90423145572662840989738057442, −8.99484395448606557115425680033, −8.328929757952894590836073374995, −7.48087799352299234132503312610, −6.91922542664631791800454552099, −6.14835192939825476313138066011, −5.63099406260341137474374777717, −4.83168211684179029828259003856, −4.04805012180908563976104756281, −2.62729617666118483826137134405, −1.92446109046378865165352183084, −0.8135157494279403059038352802, −0.11527177938891493668339716305,
1.19787779158331603191517735393, 1.78217958449013216933311887867, 2.78248833236363091745376024875, 3.85193316800731143344819702899, 4.41584526772757459339275321797, 5.372940582672227652494550461373, 6.35838603593837553482320341203, 6.8146413835962275058577445870, 7.59311236632315466151732180412, 8.388961737901347208575695404297, 9.287282858627696859383717613884, 9.84362550097226268240932870411, 10.41199604371318778586201511819, 11.30133635961620499607817849578, 11.695300930685202579194861700287, 12.20291063836139294953221329230, 13.08415877828095513229022090809, 13.72943200642657165149180954874, 14.843030472029492006334187451755, 15.672198557868666384629714331363, 16.284170139220314741007830156461, 16.72598979192935160680097670951, 17.41462980303697141298667939241, 18.05353564300444223799095248249, 18.597332896358659870788044236069