L(s) = 1 | + (1.94 + 2.05i)2-s + (−2.12 + 2.12i)3-s + (−0.408 + 7.98i)4-s + (2.24 + 2.24i)5-s + (−8.48 − 0.216i)6-s + 9.00i·7-s + (−17.1 + 14.7i)8-s − 8.99i·9-s + (−0.229 + 8.96i)10-s + (11.0 + 11.0i)11-s + (−16.0 − 17.8i)12-s + (54.5 − 54.5i)13-s + (−18.4 + 17.5i)14-s − 9.51·15-s + (−63.6 − 6.53i)16-s + 44.0·17-s + ⋯ |
L(s) = 1 | + (0.688 + 0.724i)2-s + (−0.408 + 0.408i)3-s + (−0.0510 + 0.998i)4-s + (0.200 + 0.200i)5-s + (−0.577 − 0.0147i)6-s + 0.486i·7-s + (−0.759 + 0.650i)8-s − 0.333i·9-s + (−0.00724 + 0.283i)10-s + (0.302 + 0.302i)11-s + (−0.386 − 0.428i)12-s + (1.16 − 1.16i)13-s + (−0.352 + 0.334i)14-s − 0.163·15-s + (−0.994 − 0.102i)16-s + 0.627·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.01123 + 1.35773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01123 + 1.35773i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.94 - 2.05i)T \) |
| 3 | \( 1 + (2.12 - 2.12i)T \) |
good | 5 | \( 1 + (-2.24 - 2.24i)T + 125iT^{2} \) |
| 7 | \( 1 - 9.00iT - 343T^{2} \) |
| 11 | \( 1 + (-11.0 - 11.0i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-54.5 + 54.5i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 44.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-49.9 + 49.9i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 117. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-40.6 + 40.6i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (248. + 248. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 457. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-204. - 204. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (138. + 138. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (263. + 263. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-29.1 + 29.1i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-508. + 508. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 788. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 92.2iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-914. + 914. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.45e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 229.T + 9.12e5T^{2} \) |
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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
−15.55630122657928266305738045752, −14.54113032409716800675042627098, −13.31874444102518935172405268532, −12.19886447594960233228278982510, −11.00085223287496421810059280079, −9.360242629972588664899270521275, −7.88829325974296723628293779497, −6.26961358005358694430456440400, −5.22689901345200853508595048627, −3.41698139568883987901826542814,
1.39464145672567039113512084883, 3.81008102065789634802619550318, 5.52922536228360921459003248683, 6.83595206513864537891308879022, 8.937290976993414032173219287104, 10.42749392678107716724305114305, 11.45668563753943357228887468299, 12.46922284560765949372343079375, 13.65177477498583057011143310910, 14.30265357571199789441564558077