| L(s) = 1 | + (−2.70 + 0.794i)3-s + (−0.841 − 0.540i)5-s + (−0.171 − 1.19i)7-s + (4.16 − 2.67i)9-s + (2.10 − 4.60i)11-s + (−0.740 + 5.15i)13-s + (2.70 + 0.794i)15-s + (3.58 + 4.13i)17-s + (−4.35 + 5.02i)19-s + (1.41 + 3.09i)21-s + (−2.57 + 4.04i)23-s + (0.415 + 0.909i)25-s + (−3.61 + 4.16i)27-s + (0.744 + 0.859i)29-s + (0.773 + 0.227i)31-s + ⋯ |
| L(s) = 1 | + (−1.56 + 0.458i)3-s + (−0.376 − 0.241i)5-s + (−0.0647 − 0.450i)7-s + (1.38 − 0.892i)9-s + (0.634 − 1.38i)11-s + (−0.205 + 1.42i)13-s + (0.698 + 0.205i)15-s + (0.868 + 1.00i)17-s + (−0.998 + 1.15i)19-s + (0.307 + 0.674i)21-s + (−0.536 + 0.843i)23-s + (0.0830 + 0.181i)25-s + (−0.694 + 0.801i)27-s + (0.138 + 0.159i)29-s + (0.138 + 0.0407i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.519676 + 0.380948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.519676 + 0.380948i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (2.57 - 4.04i)T \) |
| good | 3 | \( 1 + (2.70 - 0.794i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (0.171 + 1.19i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.10 + 4.60i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.740 - 5.15i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.58 - 4.13i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (4.35 - 5.02i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.744 - 0.859i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.773 - 0.227i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.750 + 0.482i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-9.50 - 6.10i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-5.84 + 1.71i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 + (-1.33 - 9.27i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.929 - 6.46i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (11.9 + 3.51i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.840 + 1.84i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-5.44 - 11.9i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.78 + 2.06i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.17 + 8.15i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (10.6 - 6.82i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-6.89 + 2.02i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (9.54 + 6.13i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.17127387698865991594174794936, −10.63201314460616709683017510103, −9.659404863273055976898478432016, −8.607339455184798778820266819264, −7.42926230819167183302397783632, −6.11902964350326590348984947695, −5.89405169462483668383820699027, −4.34983070615617072697670117329, −3.83207360860035245411481127042, −1.20800157334687812995488367985,
0.57794769407956866658201996469, 2.49680487914687996509795558359, 4.36987223328844927930240647274, 5.24545867439600923560766899860, 6.20067770669044175339240388982, 7.06131907827279079731138562832, 7.78739323410890221160157027663, 9.249033701190593124308379375377, 10.29039963791393112835767303655, 10.93319315461896891994731287663
