L(s) = 1 | + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (0.696 + 0.717i)5-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (−0.0922 − 0.995i)10-s + (0.779 + 0.626i)11-s + (−0.998 − 0.0615i)13-s + (−0.952 + 0.303i)14-s + (−0.908 + 0.417i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (−0.552 + 0.833i)20-s + (−0.213 − 0.976i)22-s + (−0.779 + 0.626i)23-s + ⋯ |
L(s) = 1 | + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (0.696 + 0.717i)5-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (−0.0922 − 0.995i)10-s + (0.779 + 0.626i)11-s + (−0.998 − 0.0615i)13-s + (−0.952 + 0.303i)14-s + (−0.908 + 0.417i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (−0.552 + 0.833i)20-s + (−0.213 − 0.976i)22-s + (−0.779 + 0.626i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.555220477 + 0.02615960306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555220477 + 0.02615960306i\) |
\(L(1)\) |
\(\approx\) |
\(0.8717277568 - 0.1227832547i\) |
\(L(1)\) |
\(\approx\) |
\(0.8717277568 - 0.1227832547i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (-0.779 - 0.626i)T \) |
| 5 | \( 1 + (0.696 + 0.717i)T \) |
| 7 | \( 1 + (0.552 - 0.833i)T \) |
| 11 | \( 1 + (0.779 + 0.626i)T \) |
| 13 | \( 1 + (-0.998 - 0.0615i)T \) |
| 17 | \( 1 + (-0.850 - 0.526i)T \) |
| 19 | \( 1 + (-0.982 - 0.183i)T \) |
| 23 | \( 1 + (-0.779 + 0.626i)T \) |
| 29 | \( 1 + (0.969 - 0.243i)T \) |
| 31 | \( 1 + (-0.816 - 0.577i)T \) |
| 37 | \( 1 + (0.602 - 0.798i)T \) |
| 41 | \( 1 + (0.969 + 0.243i)T \) |
| 43 | \( 1 + (0.389 - 0.920i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.982 + 0.183i)T \) |
| 59 | \( 1 + (0.552 + 0.833i)T \) |
| 61 | \( 1 + (-0.0307 + 0.999i)T \) |
| 67 | \( 1 + (-0.552 - 0.833i)T \) |
| 71 | \( 1 + (0.273 + 0.961i)T \) |
| 73 | \( 1 + (0.273 + 0.961i)T \) |
| 79 | \( 1 + (0.969 - 0.243i)T \) |
| 83 | \( 1 + (0.552 - 0.833i)T \) |
| 89 | \( 1 + (-0.739 - 0.673i)T \) |
| 97 | \( 1 + (0.881 + 0.473i)T \) |
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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
−21.73109787168661004177638314287, −20.845511992586872406934430369756, −19.783285456826492710006319470136, −19.379832829727335605235691603848, −18.21151668687103485880937950077, −17.71868766515180167975569770011, −16.90665576123984241327088467278, −16.387889155767653158376326560981, −15.35465668554783251960044198904, −14.555569200791553741883007289247, −14.015772778004211624468376178058, −12.77665645915860222347938914123, −11.97669376968796909067864496083, −10.970901108049905548648021983642, −10.08747400178401333274277673865, −9.14414002012578842291449562940, −8.66311712749426098508458804115, −7.97955082079705167503949253005, −6.58489274504034370628249982037, −6.0733683095328217054258430462, −5.115603879240597297849247609962, −4.35260640065859640059847678741, −2.392548120841833936321834679835, −1.78579222207376216519792615458, −0.58989350458511217587118243739,
0.740411760487409706391576967578, 2.00347215626170982819592767898, 2.46636374098908687639813168091, 3.89375396343439441019045739290, 4.5595062245943541424614966305, 6.10452713299605142731882534261, 7.18111223008552868450909414641, 7.45458984489207022455741867205, 8.809963895421612329506242713948, 9.596920515691329100097588116919, 10.26674944919314528501161982433, 10.99929998370360117428801426193, 11.73743246067668969725906365945, 12.71416289406897201460463980475, 13.63119501262779728838870332074, 14.3976863946783714616394261568, 15.2321055823886703019926180742, 16.45835302378751672637653371612, 17.418966519166846069097410654042, 17.53984023201399127308657074676, 18.36329809265039314753873097406, 19.57045126536279922647630254785, 19.82264011423839062586981492655, 20.81699268771488860212452656027, 21.580139119864833647473670813626