L(s) = 1 | + (0.891 − 0.452i)2-s + (0.591 − 0.806i)4-s + (0.101 − 0.994i)5-s + (−0.933 − 0.359i)7-s + (0.162 − 0.986i)8-s + (−0.359 − 0.933i)10-s + (0.639 − 0.768i)11-s + (−0.557 − 0.830i)13-s + (−0.994 + 0.101i)14-s + (−0.301 − 0.953i)16-s + (−0.755 − 0.654i)17-s + (0.806 + 0.591i)19-s + (−0.742 − 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (−0.872 − 0.488i)26-s + ⋯ |
L(s) = 1 | + (0.891 − 0.452i)2-s + (0.591 − 0.806i)4-s + (0.101 − 0.994i)5-s + (−0.933 − 0.359i)7-s + (0.162 − 0.986i)8-s + (−0.359 − 0.933i)10-s + (0.639 − 0.768i)11-s + (−0.557 − 0.830i)13-s + (−0.994 + 0.101i)14-s + (−0.301 − 0.953i)16-s + (−0.755 − 0.654i)17-s + (0.806 + 0.591i)19-s + (−0.742 − 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (−0.872 − 0.488i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2054510022 - 2.127502162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2054510022 - 2.127502162i\) |
\(L(1)\) |
\(\approx\) |
\(1.106080132 - 1.087287621i\) |
\(L(1)\) |
\(\approx\) |
\(1.106080132 - 1.087287621i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.891 - 0.452i)T \) |
| 5 | \( 1 + (0.101 - 0.994i)T \) |
| 7 | \( 1 + (-0.933 - 0.359i)T \) |
| 11 | \( 1 + (0.639 - 0.768i)T \) |
| 13 | \( 1 + (-0.557 - 0.830i)T \) |
| 17 | \( 1 + (-0.755 - 0.654i)T \) |
| 19 | \( 1 + (0.806 + 0.591i)T \) |
| 31 | \( 1 + (0.925 + 0.377i)T \) |
| 37 | \( 1 + (0.852 - 0.523i)T \) |
| 41 | \( 1 + (-0.540 - 0.841i)T \) |
| 43 | \( 1 + (-0.925 + 0.377i)T \) |
| 47 | \( 1 + (0.433 + 0.900i)T \) |
| 53 | \( 1 + (-0.685 - 0.728i)T \) |
| 59 | \( 1 + (0.142 + 0.989i)T \) |
| 61 | \( 1 + (-0.998 + 0.0611i)T \) |
| 67 | \( 1 + (0.768 - 0.639i)T \) |
| 71 | \( 1 + (0.0203 + 0.999i)T \) |
| 73 | \( 1 + (0.983 + 0.182i)T \) |
| 79 | \( 1 + (0.953 + 0.301i)T \) |
| 83 | \( 1 + (-0.339 + 0.940i)T \) |
| 89 | \( 1 + (-0.574 - 0.818i)T \) |
| 97 | \( 1 + (0.699 - 0.714i)T \) |
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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
−20.227604971557640190308236918246, −19.76171345816840561903719509849, −18.931420315280728213354922082963, −18.080645863695168505357849160093, −17.23665509281043631112639220039, −16.67947899615332145013438954018, −15.63295086300259832349157173700, −15.20362450611242892690624013220, −14.593003180893413003961977320664, −13.70437670930319846797080012427, −13.23307806544357535812713477507, −12.157272280772794948115013457542, −11.76575774419342091423770170050, −10.88678588266666022931201117800, −9.83922490399827893902412504287, −9.267194751029734089183389340247, −8.10186604898958069302734281713, −7.125576145686682611845416369582, −6.60520946780906470408013344373, −6.20478099202816724810718581671, −5.02126821064102496357977214210, −4.20874080682940489428578418543, −3.37939774681421059354302660702, −2.597278668755954029485251416655, −1.87137921274429328444951993710,
0.51762547912801076762310296745, 1.301177498514494822969263686049, 2.56632721797449202138005794303, 3.33462388114367086070726294266, 4.11927470218361037947580467592, 4.96837998992276903823273106237, 5.71124647033209507263839069427, 6.4233926195501488480471433334, 7.307286834413205445177121310131, 8.349163978815704472809073385208, 9.43835279821770123712240533644, 9.79529076530907729981371505748, 10.774291303889940019737651461827, 11.64609262725206151498401166059, 12.33917915282571435657928170098, 12.89075212207956524893092890031, 13.70620361649006553538074399509, 14.03895238522623164474944039878, 15.23554036479046046121821105977, 15.911449554920915603983805296416, 16.457909507599746966677334155259, 17.18358441362788877654211748959, 18.23062322302855510898230605257, 19.216181867088840340217077331444, 19.84498879653018338662398212643