| L(s) = 1 | + 0.646·5-s − i·7-s + 3.09i·11-s + 0.913i·13-s + 6.77i·17-s + 5.29·19-s − 2.53·23-s − 4.58·25-s − 9.93·29-s + 9.16i·31-s − 0.646i·35-s − 7.84i·37-s − 9.01i·41-s − 12.2·43-s + 5.65·47-s + ⋯ |
| L(s) = 1 | + 0.288·5-s − 0.377i·7-s + 0.933i·11-s + 0.253i·13-s + 1.64i·17-s + 1.21·19-s − 0.528·23-s − 0.916·25-s − 1.84·29-s + 1.64i·31-s − 0.109i·35-s − 1.28i·37-s − 1.40i·41-s − 1.86·43-s + 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8915334205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8915334205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.646T + 5T^{2} \) |
| 11 | \( 1 - 3.09iT - 11T^{2} \) |
| 13 | \( 1 - 0.913iT - 13T^{2} \) |
| 17 | \( 1 - 6.77iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 + 9.93T + 29T^{2} \) |
| 31 | \( 1 - 9.16iT - 31T^{2} \) |
| 37 | \( 1 + 7.84iT - 37T^{2} \) |
| 41 | \( 1 + 9.01iT - 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 0.417T + 73T^{2} \) |
| 79 | \( 1 + 10.7iT - 79T^{2} \) |
| 83 | \( 1 - 15.9iT - 83T^{2} \) |
| 89 | \( 1 - 9.01iT - 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.782498143149273553243679451228, −7.86196699635700122206827002123, −7.32119160625194136875341226327, −6.58886655334502723494313006794, −5.69152882905237891345878663152, −5.11487223460716398355451331542, −3.96529367929462685086313269883, −3.57517809076743854991568371368, −2.10415685097924853108735394223, −1.52320737432118670071017131966,
0.24043438150772755665033707084, 1.54544819595916813189407937594, 2.70622735508539726818594871593, 3.33283641805455264744987512793, 4.38377751658160677626533512866, 5.44267034793239321672339330919, 5.71322756487132142364775883511, 6.65595810609423923837032819106, 7.57531078447517884682299084329, 8.035504258049253101907280762742
