| L(s) = 1 | + (0.183 − 0.110i)2-s + (0.370 + 0.928i)3-s + (−0.915 + 1.72i)4-s + (−4.31 + 0.469i)5-s + (0.170 + 0.129i)6-s + (2.04 − 0.690i)7-s + (0.0458 + 0.844i)8-s + (−0.725 + 0.687i)9-s + (−0.739 + 0.562i)10-s + (−2.75 + 4.06i)11-s + (−1.94 − 0.211i)12-s + (0.125 + 0.118i)13-s + (0.299 − 0.352i)14-s + (−2.03 − 3.83i)15-s + (−2.09 − 3.08i)16-s + (6.32 + 2.13i)17-s + ⋯ |
| L(s) = 1 | + (0.129 − 0.0780i)2-s + (0.213 + 0.536i)3-s + (−0.457 + 0.863i)4-s + (−1.93 + 0.209i)5-s + (0.0695 + 0.0528i)6-s + (0.774 − 0.260i)7-s + (0.0161 + 0.298i)8-s + (−0.241 + 0.229i)9-s + (−0.233 + 0.177i)10-s + (−0.830 + 1.22i)11-s + (−0.560 − 0.0609i)12-s + (0.0348 + 0.0329i)13-s + (0.0800 − 0.0942i)14-s + (−0.525 − 0.990i)15-s + (−0.522 − 0.771i)16-s + (1.53 + 0.516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.381134 + 0.692801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.381134 + 0.692801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.370 - 0.928i)T \) |
| 59 | \( 1 + (3.76 + 6.69i)T \) |
| good | 2 | \( 1 + (-0.183 + 0.110i)T + (0.936 - 1.76i)T^{2} \) |
| 5 | \( 1 + (4.31 - 0.469i)T + (4.88 - 1.07i)T^{2} \) |
| 7 | \( 1 + (-2.04 + 0.690i)T + (5.57 - 4.23i)T^{2} \) |
| 11 | \( 1 + (2.75 - 4.06i)T + (-4.07 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-0.125 - 0.118i)T + (0.703 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-6.32 - 2.13i)T + (13.5 + 10.2i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 1.88i)T + (-18.0 + 6.06i)T^{2} \) |
| 23 | \( 1 + (0.787 + 2.83i)T + (-19.7 + 11.8i)T^{2} \) |
| 29 | \( 1 + (-2.05 - 1.23i)T + (13.5 + 25.6i)T^{2} \) |
| 31 | \( 1 + (0.933 - 5.69i)T + (-29.3 - 9.89i)T^{2} \) |
| 37 | \( 1 + (-0.237 + 4.38i)T + (-36.7 - 4.00i)T^{2} \) |
| 41 | \( 1 + (0.159 - 0.576i)T + (-35.1 - 21.1i)T^{2} \) |
| 43 | \( 1 + (-2.72 - 4.02i)T + (-15.9 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-10.0 - 1.09i)T + (45.9 + 10.1i)T^{2} \) |
| 53 | \( 1 + (5.89 + 4.48i)T + (14.1 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.62 - 4.58i)T + (28.5 - 53.8i)T^{2} \) |
| 67 | \( 1 + (-0.298 - 5.51i)T + (-66.6 + 7.24i)T^{2} \) |
| 71 | \( 1 + (1.94 + 0.211i)T + (69.3 + 15.2i)T^{2} \) |
| 73 | \( 1 + (8.10 - 9.54i)T + (-11.8 - 72.0i)T^{2} \) |
| 79 | \( 1 + (-2.36 + 5.94i)T + (-57.3 - 54.3i)T^{2} \) |
| 83 | \( 1 + (-5.01 + 2.31i)T + (53.7 - 63.2i)T^{2} \) |
| 89 | \( 1 + (-10.0 - 6.07i)T + (41.6 + 78.6i)T^{2} \) |
| 97 | \( 1 + (5.74 + 6.76i)T + (-15.6 + 95.7i)T^{2} \) |
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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
−12.53985452112498855513278507339, −12.26297698615762198161468477583, −11.14276467515586446015892886105, −10.22964352981883051351707131831, −8.660730429323468298820306029110, −7.79540748651960906021004379386, −7.46456282615215188210609941914, −4.89609563083126331749904826939, −4.17390410546525546405953018011, −3.15176245782148239380322792867,
0.72740523744736216639266599220, 3.31561709345959695416418385647, 4.71504183868088224501275195963, 5.77389139550445881826934948416, 7.52013357995012690626246786361, 8.079573848017266159306000328114, 9.035068859212605723495016354385, 10.64568964691471208393906540317, 11.48660802332973617532536484766, 12.21864499118739192708851471612
