L(s) = 1 | + (−0.860 + 0.356i)3-s + (−0.366 + 0.883i)5-s + (2.35 + 2.35i)7-s + (−1.50 + 1.50i)9-s + (1.81 + 0.752i)11-s + (−0.848 − 2.04i)13-s − 0.891i·15-s + 6.00i·17-s + (−1.54 − 3.73i)19-s + (−2.86 − 1.18i)21-s + (2.91 − 2.91i)23-s + (2.88 + 2.88i)25-s + (1.82 − 4.41i)27-s + (−9.69 + 4.01i)29-s − 7.52·31-s + ⋯ |
L(s) = 1 | + (−0.497 + 0.205i)3-s + (−0.163 + 0.395i)5-s + (0.889 + 0.889i)7-s + (−0.502 + 0.502i)9-s + (0.547 + 0.226i)11-s + (−0.235 − 0.568i)13-s − 0.230i·15-s + 1.45i·17-s + (−0.354 − 0.856i)19-s + (−0.624 − 0.258i)21-s + (0.607 − 0.607i)23-s + (0.577 + 0.577i)25-s + (0.352 − 0.850i)27-s + (−1.80 + 0.745i)29-s − 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.024784283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024784283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.860 - 0.356i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.366 - 0.883i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-2.35 - 2.35i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.81 - 0.752i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.848 + 2.04i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 6.00iT - 17T^{2} \) |
| 19 | \( 1 + (1.54 + 3.73i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.91 + 2.91i)T - 23iT^{2} \) |
| 29 | \( 1 + (9.69 - 4.01i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 + (0.994 - 2.40i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (1.37 - 1.37i)T - 41iT^{2} \) |
| 43 | \( 1 + (-10.8 - 4.50i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 3.33iT - 47T^{2} \) |
| 53 | \( 1 + (8.02 + 3.32i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (4.70 - 11.3i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (0.166 - 0.0688i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (8.39 - 3.47i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-7.92 - 7.92i)T + 71iT^{2} \) |
| 73 | \( 1 + (5.84 - 5.84i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.80iT - 79T^{2} \) |
| 83 | \( 1 + (-1.98 - 4.79i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.38 - 6.38i)T + 89iT^{2} \) |
| 97 | \( 1 - 0.874T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65817331684835556164444050019, −9.260840289157033024882024176942, −8.652798336062506862368671442100, −7.80085198068374088387092560105, −6.85867253924293807724717123995, −5.75280885564088170092281540019, −5.23086191833210990353270336753, −4.19513551417704346901068712623, −2.88147993450135134533158643994, −1.74329708134306540412101539375,
0.49881228538480653106413510910, 1.75728512786060875524401458811, 3.46200286884123333852213846787, 4.40333947855528741008815673353, 5.28736342736548576668629920087, 6.20452464741766274623489964747, 7.26617525963055363266192993019, 7.74199780461699557483294004332, 9.062956529860876129539473988787, 9.351712269324484730473534226359