| L(s) = 1 | + (0.891 + 1.48i)2-s + (2.40 + 1.79i)3-s + (0.472 − 0.892i)4-s + (−1.05 − 9.65i)5-s + (−0.523 + 5.16i)6-s + (−1.35 + 0.456i)7-s + (8.65 − 0.469i)8-s + (2.53 + 8.63i)9-s + (13.3 − 10.1i)10-s + (12.7 + 8.64i)11-s + (2.74 − 1.29i)12-s + (−9.09 − 8.61i)13-s + (−1.88 − 1.59i)14-s + (14.8 − 25.0i)15-s + (6.14 + 9.05i)16-s + (1.14 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (0.445 + 0.740i)2-s + (0.800 + 0.599i)3-s + (0.118 − 0.223i)4-s + (−0.210 − 1.93i)5-s + (−0.0873 + 0.860i)6-s + (−0.193 + 0.0651i)7-s + (1.08 − 0.0586i)8-s + (0.281 + 0.959i)9-s + (1.33 − 1.01i)10-s + (1.15 + 0.785i)11-s + (0.228 − 0.107i)12-s + (−0.699 − 0.662i)13-s + (−0.134 − 0.114i)14-s + (0.989 − 1.67i)15-s + (0.383 + 0.565i)16-s + (0.0673 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.332i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.48115 + 0.425168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48115 + 0.425168i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.40 - 1.79i)T \) |
| 59 | \( 1 + (-56.9 - 15.2i)T \) |
| good | 2 | \( 1 + (-0.891 - 1.48i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (1.05 + 9.65i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (1.35 - 0.456i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-12.7 - 8.64i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (9.09 + 8.61i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 3.39i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-3.10 - 18.9i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (24.8 - 6.90i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (-17.7 + 29.5i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (1.76 - 10.7i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-0.114 + 2.10i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (44.0 + 12.2i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (-34.3 - 50.7i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 11.2i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (14.0 - 18.4i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (57.9 - 34.8i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 21.9i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (4.68 - 43.1i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-76.5 + 90.1i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (22.9 - 57.5i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (40.3 + 87.2i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-47.3 + 78.6i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (74.8 + 88.1i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.67931407054951409153229865312, −11.86862315107474669254164044119, −10.02623404249182355628099500646, −9.516378137338841439211316948587, −8.319835365903312624173655164899, −7.54462857589368208548736931191, −5.84634873251039527584701822293, −4.78721510748799608879036346688, −4.08612772718791559900284591170, −1.61078189666687461120272899541,
2.14285521297126384589886598189, 3.15840745589559214975976934146, 3.92332335337953963257443230315, 6.58477461249447468454126993223, 6.99019594971536367085453687660, 8.139004704128806808029304060218, 9.580482816769794251931119364432, 10.69704795873052117426863992277, 11.59462786384647931095173006163, 12.19370239521791580929336045837
