| L(s) = 1 | + (0.368 + 0.170i)2-s + (−0.570 − 1.63i)3-s + (−1.18 − 1.39i)4-s + (1.50 + 2.50i)5-s + (0.0684 − 0.698i)6-s + (−0.260 − 4.81i)7-s + (−0.416 − 1.49i)8-s + (−2.34 + 1.86i)9-s + (0.128 + 1.17i)10-s + (−0.148 − 0.904i)11-s + (−1.60 + 2.74i)12-s + (2.05 − 2.70i)13-s + (0.723 − 1.81i)14-s + (3.23 − 3.89i)15-s + (−0.491 + 3.00i)16-s + (−2.51 − 0.136i)17-s + ⋯ |
| L(s) = 1 | + (0.260 + 0.120i)2-s + (−0.329 − 0.944i)3-s + (−0.594 − 0.699i)4-s + (0.673 + 1.11i)5-s + (0.0279 − 0.285i)6-s + (−0.0986 − 1.81i)7-s + (−0.147 − 0.529i)8-s + (−0.782 + 0.622i)9-s + (0.0404 + 0.372i)10-s + (−0.0447 − 0.272i)11-s + (−0.464 + 0.791i)12-s + (0.570 − 0.750i)13-s + (0.193 − 0.485i)14-s + (0.834 − 1.00i)15-s + (−0.122 + 0.750i)16-s + (−0.610 − 0.0331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0327 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0327 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.784963 - 0.759655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784963 - 0.759655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.570 + 1.63i)T \) |
| 59 | \( 1 + (7.57 + 1.25i)T \) |
| good | 2 | \( 1 + (-0.368 - 0.170i)T + (1.29 + 1.52i)T^{2} \) |
| 5 | \( 1 + (-1.50 - 2.50i)T + (-2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (0.260 + 4.81i)T + (-6.95 + 0.756i)T^{2} \) |
| 11 | \( 1 + (0.148 + 0.904i)T + (-10.4 + 3.51i)T^{2} \) |
| 13 | \( 1 + (-2.05 + 2.70i)T + (-3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (2.51 + 0.136i)T + (16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (-1.34 - 1.27i)T + (1.02 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-5.81 - 1.28i)T + (20.8 + 9.65i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.79i)T + (-18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (-2.95 - 3.12i)T + (-1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-5.07 - 1.40i)T + (31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (-1.86 - 8.49i)T + (-37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (-3.60 - 0.591i)T + (40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (11.3 + 6.81i)T + (22.0 + 41.5i)T^{2} \) |
| 53 | \( 1 + (-0.788 + 7.24i)T + (-51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (-3.36 + 7.27i)T + (-39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (-0.0835 + 0.0231i)T + (57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (-0.0441 + 0.0733i)T + (-33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (-9.58 - 3.82i)T + (52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (0.539 + 0.181i)T + (62.8 + 47.8i)T^{2} \) |
| 83 | \( 1 + (-2.93 - 4.32i)T + (-30.7 + 77.1i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 4.72i)T + (57.6 - 67.8i)T^{2} \) |
| 97 | \( 1 + (14.3 - 5.72i)T + (70.4 - 66.7i)T^{2} \) |
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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
−13.03391989926350097253972499640, −11.07881436679071546828677197936, −10.67817307244453237770701159732, −9.745110932343369385883458085373, −8.094266461767861692321967503397, −6.82908467422395945541020967936, −6.38234457635679885431238044434, −5.01525379203968748169306874282, −3.29393686356562810238050728579, −1.09070848835718902301095571985,
2.65969226273681220884409496223, 4.39335636499976927600972136704, 5.17867694911236209864002222378, 6.11212359158452249881596086253, 8.395200266263286301062864131031, 9.219070978625165511110760353100, 9.338757989291818234559618315539, 11.22720640312548521817955866425, 12.07066086126656665042243647289, 12.77587024674412153552446782889
