| 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
−11.62365268488874694913164144563, −10.90761088950754078561799483576, −10.05103431775139879391125637991, −8.623686721674459404310725614012, −7.87702009978334171145136951731, −7.19289206033700267199008827342, −5.16440054555445086763370993364, −4.26232178624134921107326202822, −1.39657334245255938382426908215, −0.07345063377660604240552211523,
3.27497301617792225114627679340, 4.76860151651763011349991498442, 6.40255222980157362701567871753, 7.22012743355082839349727996137, 8.281072932213606609000525573980, 9.181861056452167107550727429582, 10.55665559849358080990760615234, 11.03622417442030176596398807112, 11.83289297834584266443248940012, 13.02452790301848327741231206614
