L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s − i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s − i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.685427568 - 0.6030556268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.685427568 - 0.6030556268i\) |
\(L(1)\) |
\(\approx\) |
\(1.221600121 - 0.2742820743i\) |
\(L(1)\) |
\(\approx\) |
\(1.221600121 - 0.2742820743i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.382 - 0.923i)T \) |
| 13 | \( 1 + (0.923 - 0.382i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.923 - 0.382i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
| 29 | \( 1 + (-0.382 - 0.923i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.923 - 0.382i)T \) |
| 41 | \( 1 + (0.707 - 0.707i)T \) |
| 43 | \( 1 + (-0.382 + 0.923i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.382 + 0.923i)T \) |
| 59 | \( 1 + (-0.923 - 0.382i)T \) |
| 61 | \( 1 + (0.382 + 0.923i)T \) |
| 67 | \( 1 + (-0.382 - 0.923i)T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (0.923 - 0.382i)T \) |
| 89 | \( 1 + (0.707 + 0.707i)T \) |
| 97 | \( 1 - T \) |
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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
−32.58379253472138807383490546623, −31.00847310088615649815352267021, −29.86104456856111108204980774416, −28.50368400362136428091410540808, −27.94652689042179099381782483093, −26.55920035715278868897837801810, −25.70221558439378000324841207226, −24.26804127993926459196906145132, −23.070939169389033373343993813165, −21.9135506964135906894798131587, −20.83667364701691856018863980077, −20.2336018528982918517420903921, −18.061431484590094130790024917163, −17.22111079768168644084856689127, −16.28091302191520286987042906574, −14.77730978284672137861001539736, −13.77162263996464275611047970936, −12.13395465281575016471633304902, −10.70691159189086348879578866512, −9.80733575096933135088106921701, −8.48932436559246659743990461752, −6.47457800574635351897224364312, −5.097938952697780159861740463712, −3.92992400243022209039916765606, −1.5071200731640421859294593998,
1.26421235038631443804410674542, 2.772595012828704120019151091748, 5.41856088473329152163873747774, 6.23520252452283702491419180922, 7.80897195147453939782264754309, 9.177462698674183606323153391659, 10.99086787290983553485361726266, 11.869537588284717904711183926973, 13.50553301414321407586135724101, 14.10622390628662743146899280660, 15.88665217554722871348645278494, 17.45712896427119018991353075956, 18.13816302468336328779773630220, 19.05607241265494866806698771270, 20.7275066126709011622896244053, 21.9096535487864621531048276792, 22.88030335343523529783719000924, 24.42738552148706173982308687239, 24.88780613073061922891451872641, 26.14466398468374970589878207405, 27.70811869589732717816457673553, 28.729857374124470155130012114, 29.78484701874350587006825727975, 30.482273978115226370429629889285, 31.72412167262569761608345839542