L(s) = 1 | + (0.134 + 0.990i)2-s + (0.983 − 0.178i)3-s + (−0.963 + 0.266i)4-s + (−0.995 + 0.0896i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.393 − 0.919i)8-s + (0.936 − 0.351i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (0.858 + 0.512i)13-s + (0.473 − 0.880i)14-s + (−0.963 + 0.266i)15-s + (0.858 − 0.512i)16-s + (0.858 − 0.512i)17-s + (0.473 + 0.880i)18-s + ⋯ |
L(s) = 1 | + (0.134 + 0.990i)2-s + (0.983 − 0.178i)3-s + (−0.963 + 0.266i)4-s + (−0.995 + 0.0896i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.393 − 0.919i)8-s + (0.936 − 0.351i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (0.858 + 0.512i)13-s + (0.473 − 0.880i)14-s + (−0.963 + 0.266i)15-s + (0.858 − 0.512i)16-s + (0.858 − 0.512i)17-s + (0.473 + 0.880i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.240230232 + 0.8271626877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.240230232 + 0.8271626877i\) |
\(L(1)\) |
\(\approx\) |
\(1.103334272 + 0.5026583486i\) |
\(L(1)\) |
\(\approx\) |
\(1.103334272 + 0.5026583486i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (0.134 + 0.990i)T \) |
| 3 | \( 1 + (0.983 - 0.178i)T \) |
| 5 | \( 1 + (-0.995 + 0.0896i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.858 + 0.512i)T \) |
| 17 | \( 1 + (0.858 - 0.512i)T \) |
| 19 | \( 1 + (-0.0448 + 0.998i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.983 + 0.178i)T \) |
| 31 | \( 1 + (0.134 + 0.990i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.473 - 0.880i)T \) |
| 47 | \( 1 + (-0.0448 + 0.998i)T \) |
| 53 | \( 1 + (-0.995 - 0.0896i)T \) |
| 59 | \( 1 + (-0.393 + 0.919i)T \) |
| 61 | \( 1 + (0.134 - 0.990i)T \) |
| 67 | \( 1 + (0.623 - 0.781i)T \) |
| 71 | \( 1 + (0.936 + 0.351i)T \) |
| 73 | \( 1 + (0.753 - 0.657i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.134 - 0.990i)T \) |
| 89 | \( 1 + (-0.900 - 0.433i)T \) |
| 97 | \( 1 + (0.936 - 0.351i)T \) |
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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
−23.44854133773132282479019355501, −22.716262258337165605878982373990, −21.82308268834187153435175031060, −20.94908931180618073879399514062, −20.23245218118557194395028078442, −19.42990842445754856843130911167, −18.97025010754050503459474895341, −18.213862341218744938634912560724, −16.69363273833222253110863966940, −15.54518710122891995767249811987, −15.11146628594711906719110620062, −13.98588352473728780837451448820, −12.977380232622965470376016786762, −12.51851589742048659277652552841, −11.39910210749382055340682948528, −10.43560936349815112947680780976, −9.53311798872716598264260608788, −8.58796061208548097383723737131, −8.09100428018724252074005850517, −6.59647378080890022259573030045, −5.11780486555753844036774768373, −4.03614408710315347324863795330, −3.25245852037939601552573412564, −2.57460884799276133985420778125, −0.96987450638161851321620997041,
1.08586746662001087545735290645, 3.316410139746777513040195143305, 3.60615608665327113551701836630, 4.731447197582634663143718286999, 6.267744946433829489105329422549, 7.144649842903449203038253151225, 7.75941846479678745213905975602, 8.675781180315119216481381862598, 9.49455100409658123953590590972, 10.56645211115684176161314275917, 12.17556827356278178614283813616, 12.83029296030174916890801825591, 14.01773666956688540128964568343, 14.2547684414662683256911734683, 15.65287401591738455210722617294, 15.88877714408366034136195841617, 16.81265523338457486832283377056, 18.130335547708663930617378374, 19.04286147568062785714998075997, 19.38867811146395767976923695210, 20.6039190135417641910363352715, 21.39496687623997366368907846332, 22.790888442037604656126964662074, 23.247291990182091481459335743260, 23.92642213563947803591825272336