L(s) = 1 | + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.787 + 0.616i)5-s + (0.954 + 0.297i)7-s + (−0.464 + 0.885i)8-s + (−0.954 − 0.297i)10-s + (−0.787 + 0.616i)11-s + (0.748 − 0.663i)13-s + (−0.998 + 0.0603i)14-s + (0.120 − 0.992i)16-s + (0.239 − 0.970i)19-s + (0.998 − 0.0603i)20-s + (0.517 − 0.855i)22-s + (−0.707 − 0.707i)23-s + (0.239 + 0.970i)25-s + (−0.464 + 0.885i)26-s + ⋯ |
L(s) = 1 | + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.787 + 0.616i)5-s + (0.954 + 0.297i)7-s + (−0.464 + 0.885i)8-s + (−0.954 − 0.297i)10-s + (−0.787 + 0.616i)11-s + (0.748 − 0.663i)13-s + (−0.998 + 0.0603i)14-s + (0.120 − 0.992i)16-s + (0.239 − 0.970i)19-s + (0.998 − 0.0603i)20-s + (0.517 − 0.855i)22-s + (−0.707 − 0.707i)23-s + (0.239 + 0.970i)25-s + (−0.464 + 0.885i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.426350728 + 0.2738725509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.426350728 + 0.2738725509i\) |
\(L(1)\) |
\(\approx\) |
\(0.8971108430 + 0.1809605765i\) |
\(L(1)\) |
\(\approx\) |
\(0.8971108430 + 0.1809605765i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| 79 | \( 1 \) |
good | 2 | \( 1 + (-0.935 + 0.354i)T \) |
| 5 | \( 1 + (0.787 + 0.616i)T \) |
| 7 | \( 1 + (0.954 + 0.297i)T \) |
| 11 | \( 1 + (-0.787 + 0.616i)T \) |
| 13 | \( 1 + (0.748 - 0.663i)T \) |
| 19 | \( 1 + (0.239 - 0.970i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
| 29 | \( 1 + (-0.983 - 0.180i)T \) |
| 31 | \( 1 + (-0.911 - 0.410i)T \) |
| 37 | \( 1 + (0.855 - 0.517i)T \) |
| 41 | \( 1 + (0.616 - 0.787i)T \) |
| 43 | \( 1 + (-0.992 + 0.120i)T \) |
| 47 | \( 1 + (0.970 - 0.239i)T \) |
| 53 | \( 1 + (-0.464 + 0.885i)T \) |
| 59 | \( 1 + (-0.663 + 0.748i)T \) |
| 61 | \( 1 + (-0.517 - 0.855i)T \) |
| 67 | \( 1 + (-0.354 + 0.935i)T \) |
| 71 | \( 1 + (0.954 - 0.297i)T \) |
| 73 | \( 1 + (0.998 - 0.0603i)T \) |
| 83 | \( 1 + (0.663 + 0.748i)T \) |
| 89 | \( 1 + (0.885 - 0.464i)T \) |
| 97 | \( 1 + (0.855 - 0.517i)T \) |
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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
−18.363516907681466675236109707454, −17.911176393617219319185347074310, −17.10329307159813415930881530963, −16.47079827701365378949282835249, −16.15493527085354462252249475477, −15.1407876872430011601081926380, −14.22261432213824584636318526958, −13.57469937750535337350011872546, −12.94603462335813198240033673319, −12.10511334296999664666671061205, −11.347193568589844037711501732, −10.84071480274078666958538018211, −10.10950341011958044952746132504, −9.38102908026158620102165244551, −8.77293400427563564830151047389, −7.97630436408546127855976759208, −7.67290814203773855843206411015, −6.455769828495959960153909762806, −5.81121680782896041385054901650, −5.03538146548704239882877372739, −4.000183575039954668328329803, −3.257803675014744230098579020162, −2.019471459753012748226764291362, −1.68018468334105099632904969492, −0.82585190184088911362327597481,
0.69179015327368424993944021265, 1.880911080965962129558640349478, 2.23525280636799526994269411860, 3.10749637049050690853615662340, 4.44658309561966693485937553187, 5.46725180270163653649879545482, 5.75044020548646704599759307929, 6.67693036302720063270518091704, 7.54590807001224821646977584944, 7.888612975868272054899778460624, 8.892413844545760869030053133238, 9.39742667362620669916562225474, 10.26309035563412954268649798022, 10.875760449263849245905926875189, 11.1859931786785668638551728364, 12.2534624673168437287961192638, 13.139811754293476590118317915257, 13.91870287998600654364927209288, 14.62572394699930661532032440110, 15.25582799695146173302740278907, 15.64996235832541369468982468958, 16.68036412097673220503193917340, 17.32949982013193407752659108293, 18.02803102229255011751809225946, 18.28463969793462126373111192049