L(s) = 1 | + (0.618 − 0.786i)2-s + (0.866 − 0.5i)3-s + (−0.235 − 0.971i)4-s + (0.142 − 0.989i)6-s + (−0.909 − 0.415i)8-s + (0.5 − 0.866i)9-s + (−0.690 − 0.723i)12-s + (−0.281 + 0.959i)13-s + (−0.888 + 0.458i)16-s + (−0.189 + 0.981i)17-s + (−0.371 − 0.928i)18-s + (0.981 − 0.189i)19-s + (−0.458 − 0.888i)23-s + (−0.995 + 0.0950i)24-s + (0.580 + 0.814i)26-s − i·27-s + ⋯ |
L(s) = 1 | + (0.618 − 0.786i)2-s + (0.866 − 0.5i)3-s + (−0.235 − 0.971i)4-s + (0.142 − 0.989i)6-s + (−0.909 − 0.415i)8-s + (0.5 − 0.866i)9-s + (−0.690 − 0.723i)12-s + (−0.281 + 0.959i)13-s + (−0.888 + 0.458i)16-s + (−0.189 + 0.981i)17-s + (−0.371 − 0.928i)18-s + (0.981 − 0.189i)19-s + (−0.458 − 0.888i)23-s + (−0.995 + 0.0950i)24-s + (0.580 + 0.814i)26-s − i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2569804399 - 2.386735395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2569804399 - 2.386735395i\) |
\(L(1)\) |
\(\approx\) |
\(1.167575375 - 1.179192442i\) |
\(L(1)\) |
\(\approx\) |
\(1.167575375 - 1.179192442i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.618 - 0.786i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.281 + 0.959i)T \) |
| 17 | \( 1 + (-0.189 + 0.981i)T \) |
| 19 | \( 1 + (0.981 - 0.189i)T \) |
| 23 | \( 1 + (-0.458 - 0.888i)T \) |
| 29 | \( 1 + (-0.654 - 0.755i)T \) |
| 31 | \( 1 + (0.723 + 0.690i)T \) |
| 37 | \( 1 + (-0.971 - 0.235i)T \) |
| 41 | \( 1 + (0.142 - 0.989i)T \) |
| 43 | \( 1 + (-0.909 - 0.415i)T \) |
| 47 | \( 1 + (0.371 - 0.928i)T \) |
| 53 | \( 1 + (0.458 - 0.888i)T \) |
| 59 | \( 1 + (0.786 - 0.618i)T \) |
| 61 | \( 1 + (-0.928 - 0.371i)T \) |
| 67 | \( 1 + (-0.371 - 0.928i)T \) |
| 71 | \( 1 + (-0.654 - 0.755i)T \) |
| 73 | \( 1 + (-0.998 - 0.0475i)T \) |
| 79 | \( 1 + (-0.995 - 0.0950i)T \) |
| 83 | \( 1 + (-0.540 - 0.841i)T \) |
| 89 | \( 1 + (0.327 + 0.945i)T \) |
| 97 | \( 1 + (-0.909 - 0.415i)T \) |
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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
−18.55821906499114843982844602321, −18.01444227447641893358356361731, −17.22763543414358684535060366751, −16.42777670197030413456967926917, −15.8239709375288666977391324354, −15.40041840365271788660079231447, −14.6701744817461191839669081999, −14.10829599158175609655902521899, −13.40904509037618505340067086677, −13.02127686033708381876301011772, −11.98040733522161328846536125162, −11.41355781803293267103697843881, −10.280897180156183319614129735, −9.6458577620851362328743768432, −8.96773398855031990794273192269, −8.22223243193169111201650246509, −7.48753919322074623649799413582, −7.19545738995551555347469992085, −5.93268869429422024999723926678, −5.3111744857009647050907593802, −4.64954609971852474066229126743, −3.86133012912771087834252145449, −3.02366102637888845814074993396, −2.70070656198535882401783675340, −1.35391607373775026310823239853,
0.42673469152128021822657739765, 1.68873850159057955989287483033, 1.98530130832296076603320955344, 2.9368829435544421191530221446, 3.68104415250700816638866887123, 4.26777684535949707193127570455, 5.14455813484340172084284011531, 6.121787710645509389655667801592, 6.755491276180645954842378279470, 7.49637990822332208647524507762, 8.620275211375107115347051759776, 8.92738289998219836175314758261, 9.93488535088854464139222744759, 10.31178456843864471192111292317, 11.42000893478029267151318300526, 12.01186739370866831088201098991, 12.54169171445199120394693262010, 13.3295404247393392831225773326, 13.91240244376828212062792724172, 14.33061080147466205968305744075, 15.13297502624292298614365831664, 15.63218600623530835976934760214, 16.624637663497522904868552099478, 17.62616559422918277684882923164, 18.238910410596525035157250173823