| 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
−12.21864499118739192708851471612, −11.48660802332973617532536484766, −10.64568964691471208393906540317, −9.035068859212605723495016354385, −8.079573848017266159306000328114, −7.52013357995012690626246786361, −5.77389139550445881826934948416, −4.71504183868088224501275195963, −3.31561709345959695416418385647, −0.72740523744736216639266599220,
3.15176245782148239380322792867, 4.17390410546525546405953018011, 4.89609563083126331749904826939, 7.46456282615215188210609941914, 7.79540748651960906021004379386, 8.660730429323468298820306029110, 10.22964352981883051351707131831, 11.14276467515586446015892886105, 12.26297698615762198161468477583, 12.53985452112498855513278507339
