L(s) = 1 | + (0.813 + 0.581i)2-s + (0.987 − 0.154i)3-s + (0.323 + 0.946i)4-s + (−0.993 + 0.116i)5-s + (0.893 + 0.448i)6-s + (0.249 + 0.968i)7-s + (−0.286 + 0.957i)8-s + (0.952 − 0.305i)9-s + (−0.875 − 0.483i)10-s + (−0.627 − 0.778i)11-s + (0.466 + 0.884i)12-s + (0.973 + 0.230i)13-s + (−0.360 + 0.932i)14-s + (−0.963 + 0.268i)15-s + (−0.790 + 0.612i)16-s + (−0.686 − 0.727i)17-s + ⋯ |
L(s) = 1 | + (0.813 + 0.581i)2-s + (0.987 − 0.154i)3-s + (0.323 + 0.946i)4-s + (−0.993 + 0.116i)5-s + (0.893 + 0.448i)6-s + (0.249 + 0.968i)7-s + (−0.286 + 0.957i)8-s + (0.952 − 0.305i)9-s + (−0.875 − 0.483i)10-s + (−0.627 − 0.778i)11-s + (0.466 + 0.884i)12-s + (0.973 + 0.230i)13-s + (−0.360 + 0.932i)14-s + (−0.963 + 0.268i)15-s + (−0.790 + 0.612i)16-s + (−0.686 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.657064334 + 1.188982080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.657064334 + 1.188982080i\) |
\(L(1)\) |
\(\approx\) |
\(1.660666547 + 0.7546514565i\) |
\(L(1)\) |
\(\approx\) |
\(1.660666547 + 0.7546514565i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 163 | \( 1 \) |
good | 2 | \( 1 + (0.813 + 0.581i)T \) |
| 3 | \( 1 + (0.987 - 0.154i)T \) |
| 5 | \( 1 + (-0.993 + 0.116i)T \) |
| 7 | \( 1 + (0.249 + 0.968i)T \) |
| 11 | \( 1 + (-0.627 - 0.778i)T \) |
| 13 | \( 1 + (0.973 + 0.230i)T \) |
| 17 | \( 1 + (-0.686 - 0.727i)T \) |
| 19 | \( 1 + (-0.565 + 0.824i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.910 - 0.413i)T \) |
| 31 | \( 1 + (0.396 - 0.918i)T \) |
| 37 | \( 1 + (-0.0581 + 0.998i)T \) |
| 41 | \( 1 + (0.323 - 0.946i)T \) |
| 43 | \( 1 + (0.0193 - 0.999i)T \) |
| 47 | \( 1 + (0.0968 - 0.995i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.286 - 0.957i)T \) |
| 67 | \( 1 + (-0.981 + 0.192i)T \) |
| 71 | \( 1 + (0.856 + 0.516i)T \) |
| 73 | \( 1 + (-0.981 - 0.192i)T \) |
| 79 | \( 1 + (-0.740 + 0.672i)T \) |
| 83 | \( 1 + (0.713 - 0.700i)T \) |
| 89 | \( 1 + (-0.627 + 0.778i)T \) |
| 97 | \( 1 + (-0.211 - 0.977i)T \) |
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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
−27.65436755716210066484491559205, −26.594632667042162661206912130671, −25.65451539822285451358343560935, −24.322674230297318766148789160307, −23.56195074515208020349801414566, −22.87176510069072688453336999071, −21.385394254113158361747002439048, −20.644865163487097378631535396142, −19.85219713900470823562858291695, −19.30475449621951796691347960741, −17.94363127197436748777539506996, −16.102047359988522008847337389295, −15.30587577547375867760856882514, −14.57041303026561990635628187380, −13.238630383699708666879254884087, −12.84560907231847351898872005922, −11.13898654579482098669669230502, −10.55354901976152853199214810443, −9.10611157498460011167284121373, −7.81793495856323317510333407139, −6.82910168738239075640713032994, −4.780231088362939320260261276803, −4.04599701770390655889490714312, −3.04977015571685984813439544024, −1.48447287385163456590284907653,
2.38742342485597814079256272226, 3.417059331266548373134262229703, 4.48409125445556408033142821271, 5.97042073847483248828804730726, 7.24548647077303685229080162225, 8.33103805837217075321394824618, 8.79574599896063736771820958263, 10.98298045288289965874563021845, 12.00755107584405855472829979073, 13.0667351533251429492680835099, 13.96709475498019345175966577589, 15.183993815086652748765028775608, 15.507510051537552415359830020461, 16.57333643175077034299934895731, 18.46428175651768648013646249224, 18.89173105183024333358430994384, 20.48439763393742350488916944774, 20.99351489276634868596822534228, 22.164241107130131708652585852144, 23.252473566337377095599497924012, 24.21556439703914303137181003576, 24.81426127478754183993699697295, 25.89577725975096021027122813194, 26.662157194974476072650148807686, 27.61431631005072731123308632025