L(s) = 1 | + i·3-s + i·7-s − 9-s − 11-s + i·13-s + i·17-s + 19-s − 21-s − i·23-s − i·27-s + 29-s + 31-s − i·33-s − i·37-s − 39-s + ⋯ |
L(s) = 1 | + i·3-s + i·7-s − 9-s − 11-s + i·13-s + i·17-s + 19-s − 21-s − i·23-s − i·27-s + 29-s + 31-s − i·33-s − i·37-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5840185059 + 1.047498216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5840185059 + 1.047498216i\) |
\(L(1)\) |
\(\approx\) |
\(0.8450865538 + 0.5222922137i\) |
\(L(1)\) |
\(\approx\) |
\(0.8450865538 + 0.5222922137i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 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
−34.512665885422244155468985822560, −33.368390162792623722285111405705, −31.88487219300669509430517189192, −30.720036159678105293485643075963, −29.66415484553858424483262837154, −28.83127379121171043186545706856, −27.174906699397835017340244911219, −25.90667840123069701772422442237, −24.722920665161044269632960455613, −23.53597150240823310138026442521, −22.69999016720834375730312406956, −20.695607168607225609497625932771, −19.74478701066218196150919767223, −18.29799312238130571037906726382, −17.38576376059781498840809763208, −15.79427994969970989519583146281, −13.94588662492370748761203109124, −13.141235995668734664727808836599, −11.633914964105169352606859301659, −10.13039025039574289642412239789, −8.06518186518042676061520118383, −7.097471960447107766621899830927, −5.31837883635310062185777995034, −2.96145731604313918366439218618, −0.78175758422594404739951629151,
2.712571261673437121457390540174, 4.5862074698134147263949347033, 5.98776315237030887203366066327, 8.29247179462172609731476339067, 9.5392668092564573127924385132, 10.909635654089499426674194736919, 12.314118211724954003592304568031, 14.179957072571212635060024343162, 15.44477138266542007247681781931, 16.35256477930716606495463617653, 17.95182105004652510585130909180, 19.35445000819576732156807553178, 20.95428797352835588493580628252, 21.66654677955369228655182343419, 22.93786179582473127180677905639, 24.44019338983484927528478730337, 25.92992580013884052785017865113, 26.74136643289033570614674792378, 28.294360278563347705201931249151, 28.75483491129007218737184032280, 30.87016611756245345183516519694, 31.67062597000032905452552710638, 32.86231382272903645228213911845, 34.00593978863417924291162844420, 34.889410722913579019042779215255