| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.464993415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464993415\) |
| \(L(1)\) |
\(\approx\) |
\(1.472636231\) |
| \(L(1)\) |
\(\approx\) |
\(1.472636231\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 223 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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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
−26.37509942629355252909550247959, −24.72173454086261772054807864270, −24.092696934509840183417447200, −23.42920548342989780809014594121, −22.71282501605105012182620959858, −21.68439407622627582492333595530, −20.92074782965374076910003602334, −19.88064961665514151934948620314, −18.69660945795701240684169713477, −17.62087670558639267913234909493, −16.44547111963568885403485758528, −15.746878677582699078864533923462, −14.84649907114302535880416329160, −13.768038718768218694202662652802, −12.22832983769059002116508404495, −12.06956705518989711849303084799, −10.9880352542596163139014869172, −10.141230346753671663114249044395, −7.770905038828123675913917626387, −7.48148812831593664841234176241, −5.87781147248886605001186230383, −4.94605955248297894238063832415, −4.255346747726311367444444841768, −2.68415532397936651633429419649, −0.95102302934102895284307153417,
0.95102302934102895284307153417, 2.68415532397936651633429419649, 4.255346747726311367444444841768, 4.94605955248297894238063832415, 5.87781147248886605001186230383, 7.48148812831593664841234176241, 7.770905038828123675913917626387, 10.141230346753671663114249044395, 10.9880352542596163139014869172, 12.06956705518989711849303084799, 12.22832983769059002116508404495, 13.768038718768218694202662652802, 14.84649907114302535880416329160, 15.746878677582699078864533923462, 16.44547111963568885403485758528, 17.62087670558639267913234909493, 18.69660945795701240684169713477, 19.88064961665514151934948620314, 20.92074782965374076910003602334, 21.68439407622627582492333595530, 22.71282501605105012182620959858, 23.42920548342989780809014594121, 24.092696934509840183417447200, 24.72173454086261772054807864270, 26.37509942629355252909550247959
