| L(s) = 1 | + (−1.15 − 1.28i)3-s + (−0.152 + 0.417i)5-s + (−2.02 − 0.357i)7-s + (−0.315 + 2.98i)9-s + (−1.69 + 0.618i)11-s + (−1.30 + 1.09i)13-s + (0.714 − 0.288i)15-s + (−1.80 + 1.03i)17-s + (−2.98 − 1.72i)19-s + (1.88 + 3.02i)21-s + (1.25 + 7.09i)23-s + (3.67 + 3.08i)25-s + (4.20 − 3.05i)27-s + (−5.04 + 6.01i)29-s + (−4.88 + 0.861i)31-s + ⋯ |
| L(s) = 1 | + (−0.668 − 0.743i)3-s + (−0.0680 + 0.186i)5-s + (−0.766 − 0.135i)7-s + (−0.105 + 0.994i)9-s + (−0.512 + 0.186i)11-s + (−0.363 + 0.304i)13-s + (0.184 − 0.0744i)15-s + (−0.436 + 0.252i)17-s + (−0.685 − 0.395i)19-s + (0.412 + 0.659i)21-s + (0.260 + 1.47i)23-s + (0.735 + 0.617i)25-s + (0.809 − 0.587i)27-s + (−0.937 + 1.11i)29-s + (−0.877 + 0.154i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.131233 + 0.239788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131233 + 0.239788i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.15 + 1.28i)T \) |
| good | 5 | \( 1 + (0.152 - 0.417i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.02 + 0.357i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.69 - 0.618i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.30 - 1.09i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.80 - 1.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.98 + 1.72i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.25 - 7.09i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.04 - 6.01i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.88 - 0.861i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.34 + 5.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.844 - 1.00i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.952 + 2.61i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.10 - 6.27i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 4.42iT - 53T^{2} \) |
| 59 | \( 1 + (-1.57 - 0.572i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.506 + 2.87i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.09 + 7.25i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.39 - 7.60i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.57 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.55 - 9.00i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.93 + 3.29i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (14.5 + 8.39i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 4.13i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.27125340904454260455398413764, −10.86376649863733564093121818226, −9.719254500621292839873632049958, −8.741274076129106850234561007506, −7.37302431809155995384320204634, −6.98629992667752924093773722399, −5.85645913580419198010842809783, −4.91614346148895391046409094436, −3.36676475236505273485871479435, −1.88319619568276635535900083514,
0.17730376775296401057761052245, 2.71989410550180390732357171469, 4.03195510759685549287140436891, 5.02395143515653442515744661142, 6.04501602228464438398232548843, 6.86586813571867226703126472227, 8.285486815107106100815796309766, 9.175859450252098486049653815518, 10.10936649054609458819519694385, 10.68759587642429746933336743529
