| L(s) = 1 | + (0.342 + 0.939i)5-s + (−0.342 + 0.939i)11-s + (0.642 − 0.766i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.642 − 0.766i)29-s + (0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.173 − 0.984i)47-s − i·53-s − 55-s + ⋯ |
| L(s) = 1 | + (0.342 + 0.939i)5-s + (−0.342 + 0.939i)11-s + (0.642 − 0.766i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.642 − 0.766i)29-s + (0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.173 − 0.984i)47-s − i·53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.130827269 - 0.2020478686i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130827269 - 0.2020478686i\) |
| \(L(1)\) |
\(\approx\) |
\(1.119200799 + 0.1593739478i\) |
| \(L(1)\) |
\(\approx\) |
\(1.119200799 + 0.1593739478i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (0.342 + 0.939i)T \) |
| 11 | \( 1 + (-0.342 + 0.939i)T \) |
| 13 | \( 1 + (0.642 - 0.766i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.642 - 0.766i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.342 - 0.939i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.984 - 0.173i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.642 - 0.766i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−18.70191628063642532223284371774, −18.295942806927006465772522577832, −17.53839211853136776002862847204, −16.46455552259797257355022870404, −16.29193897487566920627759068327, −15.7539344831898204391839247312, −14.33831609983460319929276065105, −14.1142816396962456654329659809, −13.13203629912582976897358228130, −12.79469475219905648583722136320, −11.65326780870827608161721659487, −11.32735770531236868516292346622, −10.28642573417336790290255812463, −9.527565527903707039457803502194, −8.79140819824502346018549843289, −8.3664993590522577144677661370, −7.391445280847924729989473300595, −6.45568512319039927960566634706, −5.76015914778526459800281403968, −5.01243126996330261684624610975, −4.29966874148678177500714768209, −3.34905574983103908055713059400, −2.45154923286790756487200337687, −1.39120918057784744773530394881, −0.74658675369638555137404986510,
0.42458107377577879269984055653, 1.72893113459396995009181779762, 2.343783523924428362679502885833, 3.32587842744408466671519590074, 3.957007374407686062197476662655, 5.07524128001656112365187289384, 5.83085360124645736682708304474, 6.479896421381653419823294754885, 7.42715290496566382683167172393, 7.82084638871539287799325574595, 8.9072433684695648009736298482, 9.80401544325664044402490739218, 10.209475577307735789303066517169, 11.08998422441076157197467299265, 11.59211454721681698085780730902, 12.61806602503946870240947375692, 13.402248459038290310165830456219, 13.75046607096377692197787901968, 14.87216014678489313170020713889, 15.352471114056888147078207319960, 15.713863448814029554190264317052, 17.111536517424936910211598487511, 17.416799298227256241531627025539, 18.25879101189531122452952030494, 18.62084723450067209937472476675
