L(s) = 1 | + (−0.788 − 0.615i)2-s + (0.241 + 0.970i)4-s + (0.642 + 0.766i)7-s + (0.406 − 0.913i)8-s + (0.990 + 0.139i)11-s + (−0.788 + 0.615i)13-s + (−0.0348 − 0.999i)14-s + (−0.882 + 0.469i)16-s + (0.994 − 0.104i)17-s + (−0.913 − 0.406i)19-s + (−0.694 − 0.719i)22-s + (−0.529 − 0.848i)23-s + 26-s + (−0.587 + 0.809i)28-s + (−0.559 − 0.829i)29-s + ⋯ |
L(s) = 1 | + (−0.788 − 0.615i)2-s + (0.241 + 0.970i)4-s + (0.642 + 0.766i)7-s + (0.406 − 0.913i)8-s + (0.990 + 0.139i)11-s + (−0.788 + 0.615i)13-s + (−0.0348 − 0.999i)14-s + (−0.882 + 0.469i)16-s + (0.994 − 0.104i)17-s + (−0.913 − 0.406i)19-s + (−0.694 − 0.719i)22-s + (−0.529 − 0.848i)23-s + 26-s + (−0.587 + 0.809i)28-s + (−0.559 − 0.829i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6054214858 - 0.8053694487i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6054214858 - 0.8053694487i\) |
\(L(1)\) |
\(\approx\) |
\(0.7394397939 - 0.1829865270i\) |
\(L(1)\) |
\(\approx\) |
\(0.7394397939 - 0.1829865270i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.788 - 0.615i)T \) |
| 7 | \( 1 + (0.642 + 0.766i)T \) |
| 11 | \( 1 + (0.990 + 0.139i)T \) |
| 13 | \( 1 + (-0.788 + 0.615i)T \) |
| 17 | \( 1 + (0.994 - 0.104i)T \) |
| 19 | \( 1 + (-0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.529 - 0.848i)T \) |
| 29 | \( 1 + (-0.559 - 0.829i)T \) |
| 31 | \( 1 + (-0.997 - 0.0697i)T \) |
| 37 | \( 1 + (0.207 - 0.978i)T \) |
| 41 | \( 1 + (-0.615 - 0.788i)T \) |
| 43 | \( 1 + (0.984 - 0.173i)T \) |
| 47 | \( 1 + (-0.0697 - 0.997i)T \) |
| 53 | \( 1 + (-0.587 + 0.809i)T \) |
| 59 | \( 1 + (-0.990 + 0.139i)T \) |
| 61 | \( 1 + (-0.374 - 0.927i)T \) |
| 67 | \( 1 + (0.829 + 0.559i)T \) |
| 71 | \( 1 + (0.913 - 0.406i)T \) |
| 73 | \( 1 + (0.207 + 0.978i)T \) |
| 79 | \( 1 + (-0.559 - 0.829i)T \) |
| 83 | \( 1 + (-0.275 - 0.961i)T \) |
| 89 | \( 1 + (0.978 - 0.207i)T \) |
| 97 | \( 1 + (0.898 + 0.438i)T \) |
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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
−23.02425482508318906251296445272, −22.06813126902348454673015160151, −20.97532038306892932666882054677, −20.04121465491383400595545188271, −19.56185852769681207863046749889, −18.584393379471460404963735723588, −17.7076253229154239370730884860, −16.94377262103081677376188803462, −16.60748213795515436654650593237, −15.322195330032536859307813259504, −14.516309707904263417407950285612, −14.1345290003042587712218169128, −12.78920196548322212721068405078, −11.66554101660975332619841941342, −10.82456835870300469676127533634, −10.011571558236923348716145745855, −9.23093388119507093766568819480, −8.079975508550834184459865016153, −7.58309884092508999745328623051, −6.60130100227232566369742013317, −5.62151868184379544915566791132, −4.66460158652417429997271319396, −3.475208193736602569710039347397, −1.83215316125074939878275649445, −1.02626966978025108340616095017,
0.342381356524208048694658047310, 1.77661016871699753509041864155, 2.35787554287797938945642918801, 3.736025997845919958085423450405, 4.6296122013088291963173235562, 5.97343399910351655298726772782, 7.0764629718809177346797446365, 7.934144334350261063922867374551, 8.93405528811908714537985885601, 9.42707855686992076759571132567, 10.508609096239589632531913590421, 11.426507485862881611099241042734, 12.11655350092928211192752306791, 12.68337608591902959877531222042, 14.13370981991064566554012291035, 14.79321957386923109212187443785, 15.8726248418166249923001420411, 16.98911489791586589168247621266, 17.247262880867635715984650665145, 18.47407755619728851534929554486, 18.905893942485555749959094052630, 19.80311938606603742982337316431, 20.58541702871038148211692444978, 21.5571874040709945929094894193, 21.88340934376238141356108665708