| L(s) = 1 | + (−1.65 + 0.603i)2-s + (−1.50 + 0.853i)3-s + (0.851 − 0.714i)4-s + (−3.19 + 2.67i)5-s + (1.98 − 2.32i)6-s + (0.729 + 2.54i)7-s + (0.783 − 1.35i)8-s + (1.54 − 2.57i)9-s + (3.67 − 6.36i)10-s + (−1.89 − 1.58i)11-s + (−0.673 + 1.80i)12-s + (0.108 − 0.0907i)13-s + (−2.74 − 3.77i)14-s + (2.52 − 6.75i)15-s + (−0.866 + 4.91i)16-s + (−0.351 + 0.608i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 + 0.426i)2-s + (−0.870 + 0.492i)3-s + (0.425 − 0.357i)4-s + (−1.42 + 1.19i)5-s + (0.809 − 0.948i)6-s + (0.275 + 0.961i)7-s + (0.277 − 0.479i)8-s + (0.514 − 0.857i)9-s + (1.16 − 2.01i)10-s + (−0.570 − 0.478i)11-s + (−0.194 + 0.520i)12-s + (0.0299 − 0.0251i)13-s + (−0.733 − 1.00i)14-s + (0.651 − 1.74i)15-s + (−0.216 + 1.22i)16-s + (−0.0852 + 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0900 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0900 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0314116 - 0.0343786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0314116 - 0.0343786i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 - 0.853i)T \) |
| 7 | \( 1 + (-0.729 - 2.54i)T \) |
| good | 2 | \( 1 + (1.65 - 0.603i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (3.19 - 2.67i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (1.89 + 1.58i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.108 + 0.0907i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.351 - 0.608i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.23 + 5.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.24 - 1.90i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.100 - 0.0846i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.55 - 2.98i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 - 0.775T + 37T^{2} \) |
| 41 | \( 1 + (2.52 - 2.11i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.66 - 2.06i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (4.55 + 3.82i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 3.22i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.630 + 3.57i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.167 + 0.140i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (14.4 + 5.24i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (7.04 + 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + (5.42 - 1.97i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.84 + 1.54i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.452 + 0.783i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 - 0.629i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.02452790301848327741231206614, −11.83289297834584266443248940012, −11.03622417442030176596398807112, −10.55665559849358080990760615234, −9.181861056452167107550727429582, −8.281072932213606609000525573980, −7.22012743355082839349727996137, −6.40255222980157362701567871753, −4.76860151651763011349991498442, −3.27497301617792225114627679340,
0.07345063377660604240552211523, 1.39657334245255938382426908215, 4.26232178624134921107326202822, 5.16440054555445086763370993364, 7.19289206033700267199008827342, 7.87702009978334171145136951731, 8.623686721674459404310725614012, 10.05103431775139879391125637991, 10.90761088950754078561799483576, 11.62365268488874694913164144563
