L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s − i·7-s + i·8-s + 9-s − 10-s + i·11-s + 12-s − 14-s + i·15-s + 16-s + 17-s − i·18-s + ⋯ |
L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s − i·7-s + i·8-s + 9-s − 10-s + i·11-s + 12-s − 14-s + i·15-s + 16-s + 17-s − i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2422224665 - 0.3264207507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2422224665 - 0.3264207507i\) |
\(L(1)\) |
\(\approx\) |
\(0.4095963894 - 0.4599700051i\) |
\(L(1)\) |
\(\approx\) |
\(0.4095963894 - 0.4599700051i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 79 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + iT \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - iT \) |
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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
−22.426350932102894939278033289447, −21.683934653870832963017179095483, −21.06064457271699352962418753314, −19.0848033287710941968775195331, −18.844274399212069514033400973693, −18.1721671362129215109011213513, −17.46827883091420878702128117420, −16.48240188672338096726988870229, −16.000670847587882563578147665301, −15.18568683349599901122162387451, −14.396199295243716181806161912574, −13.69590945988982764689036309585, −12.47826336942322086166432704128, −11.9458936824497306046919853407, −10.87480191272503441608571075653, −10.11718168389799132909443677441, −9.2618235922152787573784247611, −8.09892938378656262627132563586, −7.440551394663105625577683191011, −6.29284264324948038380197095171, −5.92929538928359680995942772114, −5.288691630900300330251739716561, −3.98534330776446608430545455285, −3.10703094251998038267975128830, −1.4941976717759300366027283005,
0.22699630115796122879837937187, 1.19480497410937830648454443465, 2.05848864322952149153331111713, 3.74675997964592752789786381101, 4.38506206858392515684900073822, 5.06046603313242314686161582845, 5.95348434963936241326686578599, 7.32684410452773365602749758327, 7.9979830105792325763033830774, 9.483482098921627830558693682277, 9.752560009606910514569392564411, 10.73855589518794289382755444437, 11.48012812844343068051012315751, 12.27967395067262494998635610799, 12.867632931996497860614484831895, 13.497347108459321041532302246808, 14.55047287243532380558671624958, 15.746686969430792939681525957622, 16.69289959766587258415719676778, 17.22221047624480028674222062223, 17.80904268793153432281741597020, 18.6924853392285120047315428001, 19.74145735525456860096289387875, 20.2970037199496067771388737717, 20.96123186256391004951481435137