L(s) = 1 | + (−0.751 − 0.659i)3-s + (−0.0654 + 0.997i)5-s + (0.130 + 0.991i)9-s + (0.321 − 0.946i)11-s + (0.831 − 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (0.442 − 0.896i)19-s + (−0.991 + 0.130i)23-s + (−0.991 − 0.130i)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 + (0.997 + 0.0654i)37-s + ⋯ |
L(s) = 1 | + (−0.751 − 0.659i)3-s + (−0.0654 + 0.997i)5-s + (0.130 + 0.991i)9-s + (0.321 − 0.946i)11-s + (0.831 − 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (0.442 − 0.896i)19-s + (−0.991 + 0.130i)23-s + (−0.991 − 0.130i)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 + (0.997 + 0.0654i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8848757595 - 0.4613711822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8848757595 - 0.4613711822i\) |
\(L(1)\) |
\(\approx\) |
\(0.8309916949 - 0.1380500950i\) |
\(L(1)\) |
\(\approx\) |
\(0.8309916949 - 0.1380500950i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.751 - 0.659i)T \) |
| 5 | \( 1 + (-0.0654 + 0.997i)T \) |
| 11 | \( 1 + (0.321 - 0.946i)T \) |
| 13 | \( 1 + (0.831 - 0.555i)T \) |
| 17 | \( 1 + (-0.965 + 0.258i)T \) |
| 19 | \( 1 + (0.442 - 0.896i)T \) |
| 23 | \( 1 + (-0.991 + 0.130i)T \) |
| 29 | \( 1 + (-0.980 - 0.195i)T \) |
| 31 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (0.997 + 0.0654i)T \) |
| 41 | \( 1 + (-0.382 + 0.923i)T \) |
| 43 | \( 1 + (0.195 + 0.980i)T \) |
| 47 | \( 1 + (0.258 - 0.965i)T \) |
| 53 | \( 1 + (0.321 - 0.946i)T \) |
| 59 | \( 1 + (0.896 - 0.442i)T \) |
| 61 | \( 1 + (0.946 - 0.321i)T \) |
| 67 | \( 1 + (-0.751 - 0.659i)T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 + (0.793 - 0.608i)T \) |
| 79 | \( 1 + (0.965 + 0.258i)T \) |
| 83 | \( 1 + (-0.555 - 0.831i)T \) |
| 89 | \( 1 + (0.608 - 0.793i)T \) |
| 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
−22.234294360135985974181250178445, −21.16618829059744374936309742161, −20.516419331869494422614628696741, −20.10601941695336077339349250823, −18.77839994198757553716576114873, −17.90504091197653397025512505879, −17.22218310097678503666185401446, −16.45680179315961051156532362448, −15.86339614423262029774151723849, −15.15080584053972055949992470153, −14.04052693610405461144818141058, −13.09368321804675902839078716088, −12.15834424767065461731034532668, −11.70108580704314665917517823811, −10.71470857673490800622269645014, −9.7011605388344797814608811581, −9.19511041840866695912793808183, −8.223196368776784330044132371233, −7.03786453127572304311913051646, −6.046205523961398752847171256198, −5.31438370345092735636646487195, −4.14493755848135513222952027492, −4.02797903364770454244161650440, −2.10795535014035487612380800057, −0.999501127584761548850272316,
0.60982070796597518333135799537, 1.93046548002453477722105178050, 2.97869561873117260591459106732, 3.99407770383174277503350361540, 5.28675946264742888385137187978, 6.27082615079411439026885748210, 6.5881755932403276571993626689, 7.7322947199132331775583923758, 8.432635400953551211821573112269, 9.74118671997777448518559104420, 10.78748311497862978266961808925, 11.249434701005701262591858560554, 11.86716143837480957696900681432, 13.27327352739229691601342540936, 13.47941001050862894706985017492, 14.56333520466989554633412271537, 15.58525095236884496474893718336, 16.23005241405622066853594597617, 17.268890161559743022154199248430, 18.07102641493802743623298901057, 18.39602504289260878102283610393, 19.42417360021707828096249726095, 19.92686501420987449879347747157, 21.33530153773567409873691192324, 22.090216221755631944134126976327