| L(s) = 1 | + (0.924 − 2.53i)2-s + (0.795 − 1.53i)3-s + (−4.05 − 3.40i)4-s + (−0.268 + 1.52i)5-s + (−3.17 − 3.44i)6-s + (1.75 + 1.97i)7-s + (−7.72 + 4.45i)8-s + (−1.73 − 2.44i)9-s + (3.61 + 2.08i)10-s + (3.67 − 0.647i)11-s + (−8.47 + 3.53i)12-s + (0.202 + 0.555i)13-s + (6.64 − 2.62i)14-s + (2.12 + 1.62i)15-s + (2.34 + 13.2i)16-s + (2.21 − 3.83i)17-s + ⋯ |
| L(s) = 1 | + (0.653 − 1.79i)2-s + (0.459 − 0.888i)3-s + (−2.02 − 1.70i)4-s + (−0.119 + 0.680i)5-s + (−1.29 − 1.40i)6-s + (0.663 + 0.748i)7-s + (−2.72 + 1.57i)8-s + (−0.578 − 0.816i)9-s + (1.14 + 0.659i)10-s + (1.10 − 0.195i)11-s + (−2.44 + 1.02i)12-s + (0.0560 + 0.153i)13-s + (1.77 − 0.701i)14-s + (0.548 + 0.418i)15-s + (0.585 + 3.32i)16-s + (0.537 − 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.185242 - 1.64409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185242 - 1.64409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.795 + 1.53i)T \) |
| 7 | \( 1 + (-1.75 - 1.97i)T \) |
| good | 2 | \( 1 + (-0.924 + 2.53i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.268 - 1.52i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-3.67 + 0.647i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.202 - 0.555i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.21 + 3.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.88 - 3.39i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 1.45i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.530 - 1.45i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.97 - 4.74i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.87 - 3.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.26 - 0.459i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.06 + 6.03i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.70 + 3.95i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 2.96iT - 53T^{2} \) |
| 59 | \( 1 + (-0.751 + 4.26i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.03 - 7.19i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.30 + 1.56i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.85 - 2.80i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.50 + 2.59i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.68 + 2.79i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (4.97 + 1.80i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.95 + 5.11i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (18.3 - 3.24i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.07558622787804955886178584236, −11.46686255072561951159417119986, −10.53428207821538210833767076720, −9.203181459383251182747203103172, −8.510984392092093744958243361968, −6.72862010046842386826222885758, −5.41416107799673259066159997299, −3.83608581206074044718695946246, −2.72045363985681066446481020341, −1.55556284177566682747123725013,
3.89766894190991657579219031512, 4.41564342774466769741217725353, 5.49182246141869779781011826909, 6.80022545148229834486524106650, 8.003920823802723243314823379841, 8.627733977311909672780644463453, 9.537473579814014660276852604733, 11.04904813675747390400537485573, 12.56017980456344664542161899723, 13.38727538029849744179343652791
