| L(s) = 1 | + (−0.826 + 0.563i)2-s + (−0.946 + 1.45i)3-s + (0.365 − 0.930i)4-s + (0.685 + 3.00i)5-s + (−0.0351 − 1.73i)6-s + (−1.02 − 2.43i)7-s + (0.222 + 0.974i)8-s + (−1.20 − 2.74i)9-s + (−2.25 − 2.09i)10-s + (3.74 + 1.80i)11-s + (1.00 + 1.41i)12-s + (4.64 − 3.16i)13-s + (2.22 + 1.43i)14-s + (−5.00 − 1.84i)15-s + (−0.733 − 0.680i)16-s + (1.61 + 4.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.584 + 0.398i)2-s + (−0.546 + 0.837i)3-s + (0.182 − 0.465i)4-s + (0.306 + 1.34i)5-s + (−0.0143 − 0.706i)6-s + (−0.388 − 0.921i)7-s + (0.0786 + 0.344i)8-s + (−0.402 − 0.915i)9-s + (−0.714 − 0.662i)10-s + (1.13 + 0.544i)11-s + (0.289 + 0.407i)12-s + (1.28 − 0.878i)13-s + (0.593 + 0.383i)14-s + (−1.29 − 0.477i)15-s + (−0.183 − 0.170i)16-s + (0.392 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.481162 + 0.944426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.481162 + 0.944426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 3 | \( 1 + (0.946 - 1.45i)T \) |
| 7 | \( 1 + (1.02 + 2.43i)T \) |
| good | 5 | \( 1 + (-0.685 - 3.00i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (-3.74 - 1.80i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-4.64 + 3.16i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.61 - 4.11i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-3.38 - 5.85i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.37 - 2.97i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (4.57 + 0.689i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-0.295 - 0.511i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.18 + 0.931i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (5.00 + 1.54i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (-6.48 + 2.00i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (-7.74 + 5.28i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (5.81 - 0.877i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (1.64 - 0.507i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (4.02 + 10.2i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-4.10 - 7.11i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.20 + 1.50i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.216 - 2.88i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (5.84 - 10.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.21 + 6.28i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (-13.4 - 9.16i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-1.51 - 2.62i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.39205400427499328862735984588, −9.847568253548413740680070426769, −8.961534494434548509041781363074, −7.75925081949701095055140872955, −6.89486253666724694859752313748, −6.19210845678253104599982299325, −5.54730544005438023470993432465, −3.77833867829398358178403605114, −3.49164508611530767623859492215, −1.34376843806845616492126513379,
0.793384832828634951765739288844, 1.65765143769933329785923281108, 3.05909753458744322875379117718, 4.60691064970751109896597944825, 5.59950242150970981998867237311, 6.39289777789669536300749345208, 7.26740798713401212455099358483, 8.568094505611781603206575805171, 8.994183268912659263262946468772, 9.419005274142106173838113232099
