| L(s) = 1 | + (−1.41 + 0.0847i)2-s + (−1.25 + 0.632i)3-s + (1.98 − 0.239i)4-s + (−1.02 − 2.31i)5-s + (1.72 − 1.00i)6-s + (−3.96 − 2.37i)7-s + (−2.78 + 0.506i)8-s + (−0.606 + 0.817i)9-s + (1.64 + 3.18i)10-s + (−0.763 − 0.596i)11-s + (−2.34 + 1.55i)12-s + (0.884 − 0.842i)13-s + (5.80 + 3.02i)14-s + (2.75 + 2.26i)15-s + (3.88 − 0.950i)16-s + (2.99 + 3.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 + 0.0599i)2-s + (−0.725 + 0.365i)3-s + (0.992 − 0.119i)4-s + (−0.459 − 1.03i)5-s + (0.702 − 0.408i)6-s + (−1.50 − 0.899i)7-s + (−0.983 + 0.178i)8-s + (−0.202 + 0.272i)9-s + (0.520 + 1.00i)10-s + (−0.230 − 0.179i)11-s + (−0.677 + 0.449i)12-s + (0.245 − 0.233i)13-s + (1.55 + 0.807i)14-s + (0.711 + 0.584i)15-s + (0.971 − 0.237i)16-s + (0.725 + 0.884i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.234722 + 0.180822i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.234722 + 0.180822i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0847i)T \) |
| good | 3 | \( 1 + (1.25 - 0.632i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (1.02 + 2.31i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (3.96 + 2.37i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (0.763 + 0.596i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.884 + 0.842i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 3.64i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 6.51i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (4.26 + 2.01i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-2.84 - 1.61i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-5.00 + 0.995i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-5.53 - 3.50i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.0248 + 0.0274i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.94 - 8.90i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (5.53 + 2.95i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (6.87 - 3.89i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (7.82 - 8.22i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.282 - 3.83i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-1.17 - 1.01i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (2.14 + 0.317i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-6.58 + 3.94i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (5.21 + 1.58i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-10.8 + 6.85i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-3.02 + 1.42i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (9.32 + 6.23i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.68306519239660454797090777094, −10.21884894750904935623064689499, −9.541567636594537060956278634633, −8.222559343605074809822239539170, −7.86181155477447718037485981361, −6.33380869639313978535938070908, −5.86661594513961953141091153886, −4.37926096072474365371009286222, −3.18760526858831587157658998440, −1.00158354467265755652598654521,
0.33176062685734923492672212665, 2.65075893260801112449784984941, 3.30780421944370813564785899299, 5.54759419641076609862540485036, 6.54017285766193780953149616721, 6.82314944560479704147711672795, 7.923899203818161064966630984250, 9.238240659856426920389329438174, 9.670782169528218906810856686623, 10.76748673007529662609505273480
