L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.939 − 0.342i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)14-s + (−0.642 − 0.766i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.939 − 0.342i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)14-s + (−0.642 − 0.766i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1808164543 - 0.7401300877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1808164543 - 0.7401300877i\) |
\(L(1)\) |
\(\approx\) |
\(0.6395703065 - 0.4672474606i\) |
\(L(1)\) |
\(\approx\) |
\(0.6395703065 - 0.4672474606i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (0.342 + 0.939i)T \) |
| 13 | \( 1 + (-0.766 - 0.642i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.984 + 0.173i)T \) |
| 31 | \( 1 + (0.984 + 0.173i)T \) |
| 41 | \( 1 + (0.642 + 0.766i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.642 - 0.766i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.342 - 0.939i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.642 + 0.766i)T \) |
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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
−18.84142235970849354809415105519, −18.673517646827844159468390741679, −17.136470926372490244738919556560, −16.92466372817690617019182231048, −16.05669502042867685800410746482, −15.508733867989889041955515299951, −14.799628346542549909833363600535, −14.335025375034966142204803432135, −13.7653476078493428195402182500, −12.92710620061187166735827025564, −11.84263111101649239172186550901, −10.92846767702671642502288133426, −10.32659670706537410542848160085, −9.87678137353065910735202332844, −9.03545292546157747248382962979, −8.53899008667501484352745418152, −7.48691704937747104424570983128, −7.31958065936430578237590473140, −6.3073419086393351666366006053, −5.85506242057366303060119567272, −4.48052353266301347481227607682, −3.778630532198035603313098466524, −3.03866926753796962460184505311, −2.249982640825193384611974271076, −1.229558247199084794278647562558,
0.26325560681308780525241315129, 1.13767994590661152962810324162, 2.089729843528417096481714990802, 2.85005141646583390685996097197, 3.35732277459670030663747788949, 4.41108021223099095982378743779, 4.99387392829601579656099790753, 6.440310648254974727268134305456, 7.309225995259648168633451173810, 7.53709629417755724156686081836, 8.64226345457217498189385988579, 9.02859318001415579980447401561, 9.70022682329902931078546047915, 10.02018917376978630219562036127, 11.34577241647896328716997280605, 12.04035700256911967549122359566, 12.63399539662244414869830412773, 13.02999088283004526861720841440, 13.628766887963634905947494604127, 14.92241744070447669617700432364, 15.38991245697719539939974339566, 16.07668033554885520626504112632, 16.86072308234080515632302954613, 17.4802518310173117493218777818, 18.23207073165279733772362186902