L(s) = 1 | + (0.946 − 0.321i)3-s + (0.896 − 0.442i)5-s + (0.793 − 0.608i)9-s + (0.659 + 0.751i)11-s + (0.831 − 0.555i)13-s + (0.707 − 0.707i)15-s + (0.258 − 0.965i)17-s + (−0.997 + 0.0654i)19-s + (0.608 + 0.793i)23-s + (0.608 − 0.793i)25-s + (0.555 − 0.831i)27-s + (−0.980 − 0.195i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)33-s + (−0.442 − 0.896i)37-s + ⋯ |
L(s) = 1 | + (0.946 − 0.321i)3-s + (0.896 − 0.442i)5-s + (0.793 − 0.608i)9-s + (0.659 + 0.751i)11-s + (0.831 − 0.555i)13-s + (0.707 − 0.707i)15-s + (0.258 − 0.965i)17-s + (−0.997 + 0.0654i)19-s + (0.608 + 0.793i)23-s + (0.608 − 0.793i)25-s + (0.555 − 0.831i)27-s + (−0.980 − 0.195i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)33-s + (−0.442 − 0.896i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.502303042 - 0.9243948451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.502303042 - 0.9243948451i\) |
\(L(1)\) |
\(\approx\) |
\(1.738082896 - 0.3633552460i\) |
\(L(1)\) |
\(\approx\) |
\(1.738082896 - 0.3633552460i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.946 - 0.321i)T \) |
| 5 | \( 1 + (0.896 - 0.442i)T \) |
| 11 | \( 1 + (0.659 + 0.751i)T \) |
| 13 | \( 1 + (0.831 - 0.555i)T \) |
| 17 | \( 1 + (0.258 - 0.965i)T \) |
| 19 | \( 1 + (-0.997 + 0.0654i)T \) |
| 23 | \( 1 + (0.608 + 0.793i)T \) |
| 29 | \( 1 + (-0.980 - 0.195i)T \) |
| 31 | \( 1 + (-0.866 + 0.5i)T \) |
| 37 | \( 1 + (-0.442 - 0.896i)T \) |
| 41 | \( 1 + (-0.382 + 0.923i)T \) |
| 43 | \( 1 + (0.195 + 0.980i)T \) |
| 47 | \( 1 + (-0.965 + 0.258i)T \) |
| 53 | \( 1 + (0.659 + 0.751i)T \) |
| 59 | \( 1 + (-0.0654 + 0.997i)T \) |
| 61 | \( 1 + (-0.751 - 0.659i)T \) |
| 67 | \( 1 + (0.946 - 0.321i)T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 + (0.130 + 0.991i)T \) |
| 79 | \( 1 + (-0.258 - 0.965i)T \) |
| 83 | \( 1 + (-0.555 - 0.831i)T \) |
| 89 | \( 1 + (-0.991 - 0.130i)T \) |
| 97 | \( 1 - iT \) |
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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
−21.90064909870573529570658621082, −21.16376092584698294759394979951, −20.71175078424804821308799850239, −19.592995620332568481481845253809, −18.845563201491782936367682532915, −18.42586542321883373446435487769, −16.97377131610627099844117742651, −16.7104543687874228718760835127, −15.422393346469035105650490681404, −14.71286790060036029676797971900, −14.110661000868826715102241064057, −13.358838635094565626237046386568, −12.68146580476479833848823505015, −11.18262403624390704625577245969, −10.62128833421079614693430590797, −9.71299942926400927601067740806, −8.80946473243064030206756714853, −8.42293881147252435626543359478, −7.01009289756316349979785759421, −6.33529411500309014744345505064, −5.32915935174003441148241307581, −3.98473755869870868797510580503, −3.429517193834604514362249454818, −2.21382322508426483277427728221, −1.509328340122821286827385240453,
1.2171424894652723406429639129, 1.917432016349344225386306252625, 2.995824964783464600427883757089, 3.98619893378171631166749993998, 5.05319880265286813172453213197, 6.11193966973714317581230023709, 6.995056485373670425874312391347, 7.88013303146373026259397252608, 8.94841392440342892669116781109, 9.34222465280630543994339692216, 10.21033801022576162249831747492, 11.33474504515508034269861532774, 12.57710910387916174639181122293, 13.00163254618180810501217239879, 13.8184455520296575488611121934, 14.56486546173626465958204200826, 15.28764206273827329558757653323, 16.29593694058605563074480841303, 17.21825064673802183873424511898, 18.01630044682113263530170246499, 18.59998778890968177467244923257, 19.73889817292283165682701050483, 20.219807695159396938153286793739, 21.073868308712391761719646086023, 21.50079596165144707219210181339