L(s) = 1 | + (−0.281 + 0.959i)3-s + (0.989 + 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.989 − 0.142i)13-s + (−0.755 − 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (0.755 − 0.654i)27-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.989 + 0.142i)33-s + (0.540 − 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯ |
L(s) = 1 | + (−0.281 + 0.959i)3-s + (0.989 + 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.989 − 0.142i)13-s + (−0.755 − 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (0.755 − 0.654i)27-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.989 + 0.142i)33-s + (0.540 − 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.212359034 + 0.9903414512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.212359034 + 0.9903414512i\) |
\(L(1)\) |
\(\approx\) |
\(1.062403054 + 0.4265603348i\) |
\(L(1)\) |
\(\approx\) |
\(1.062403054 + 0.4265603348i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.281 + 0.959i)T \) |
| 7 | \( 1 + (0.989 + 0.142i)T \) |
| 11 | \( 1 + (0.415 + 0.909i)T \) |
| 13 | \( 1 + (0.989 - 0.142i)T \) |
| 17 | \( 1 + (-0.755 - 0.654i)T \) |
| 19 | \( 1 + (0.654 + 0.755i)T \) |
| 29 | \( 1 + (-0.654 + 0.755i)T \) |
| 31 | \( 1 + (0.959 - 0.281i)T \) |
| 37 | \( 1 + (0.540 - 0.841i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.281 + 0.959i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.989 - 0.142i)T \) |
| 59 | \( 1 + (0.142 + 0.989i)T \) |
| 61 | \( 1 + (0.959 - 0.281i)T \) |
| 67 | \( 1 + (0.909 + 0.415i)T \) |
| 71 | \( 1 + (-0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.755 + 0.654i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (0.540 - 0.841i)T \) |
| 89 | \( 1 + (0.959 + 0.281i)T \) |
| 97 | \( 1 + (0.540 + 0.841i)T \) |
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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
−21.809544623715303706031641164514, −20.85683479168365057966160764715, −20.08045035624390546685639586393, −19.18538257318033041030349912011, −18.57442161261305693109641302789, −17.67343704849608327073833101325, −17.26204635562817102399553491379, −16.26899375798768042229271851039, −15.32217722977648466491714564203, −14.25540165677711114373416805931, −13.64673288079038654552860651331, −13.04596446640443842233216351352, −11.83213906106692960096385932077, −11.28056255421363143754811122880, −10.768598005233637544052735279555, −9.23423772172688836044508159652, −8.35373030120086467770796220109, −7.85752951537291806230422035434, −6.66991119855311810179344911348, −6.08210693631944312347929330414, −5.08595665261627514396066626019, −4.02046468149183105775017502925, −2.79777542203863071103217918503, −1.665300837262238845754051814892, −0.86928163425187815239942504859,
1.188161405987084497155414912195, 2.406667770944598223429285726268, 3.68790683656964578947275978253, 4.42684411659654268649271134942, 5.23319309140436850905048723293, 6.08029918120039030296782503545, 7.221351944389489770341386369, 8.266609036236352191729064439279, 9.08002985183481592584007374276, 9.83746595302906716824867490520, 10.80961078438090615652609633678, 11.43711085070882997988935525096, 12.11746407844104136574737433226, 13.31253241990529364151815019102, 14.3849660937884350050542092043, 14.81710685495609691972349384924, 15.79961210589764150771647197546, 16.31146149263951654797848985112, 17.50272448487275179201534399435, 17.8110375625916298678296891346, 18.7449880672098360273457494366, 20.21307194784394070520469997351, 20.41002701783380794815462457347, 21.22711721074019245897171558680, 22.017558116711365744743989599852