| L(s) = 1 | + (1.30 + 0.547i)2-s + (1.98 − 1.00i)3-s + (1.40 + 1.42i)4-s + (0.477 + 1.07i)5-s + (3.13 − 0.217i)6-s + (−2.49 − 1.49i)7-s + (1.04 + 2.62i)8-s + (1.16 − 1.56i)9-s + (0.0333 + 1.66i)10-s + (0.535 + 0.417i)11-s + (4.21 + 1.43i)12-s + (3.15 − 3.00i)13-s + (−2.43 − 3.31i)14-s + (2.02 + 1.66i)15-s + (−0.0720 + 3.99i)16-s + (−1.71 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (0.922 + 0.386i)2-s + (1.14 − 0.577i)3-s + (0.700 + 0.713i)4-s + (0.213 + 0.481i)5-s + (1.28 − 0.0887i)6-s + (−0.941 − 0.564i)7-s + (0.370 + 0.928i)8-s + (0.387 − 0.522i)9-s + (0.0105 + 0.526i)10-s + (0.161 + 0.125i)11-s + (1.21 + 0.413i)12-s + (0.875 − 0.833i)13-s + (−0.649 − 0.884i)14-s + (0.523 + 0.429i)15-s + (−0.0180 + 0.999i)16-s + (−0.415 − 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.26726 + 0.560634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.26726 + 0.560634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 - 0.547i)T \) |
| good | 3 | \( 1 + (-1.98 + 1.00i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.477 - 1.07i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (2.49 + 1.49i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.535 - 0.417i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.15 + 3.00i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.71 + 2.08i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 4.99i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (1.73 + 0.822i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (1.80 + 1.02i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (6.69 - 1.33i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (6.39 + 4.04i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (6.91 + 7.63i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-1.82 - 5.52i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (2.90 + 1.55i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (2.55 - 1.44i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-5.98 + 6.28i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.117 - 1.59i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-9.51 - 8.20i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (0.785 + 0.116i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-1.81 + 1.08i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (6.27 + 1.90i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-0.859 + 0.544i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 4.82i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 8.11i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.94232864207066813846272880318, −10.14411418210040502425487080460, −8.886917278889349303096806394917, −8.034674461380839021112424149604, −7.18208899883495926765747786973, −6.51834484274491042481070029457, −5.46597584309866749319361800897, −3.74825796494539377538151342825, −3.26004675419166628013093727624, −2.04821045313022609591212687882,
1.88795824641933370583629656505, 3.13841572093686494207725435997, 3.80687017640467967827609824441, 4.94603870307418549318735852351, 6.09420196114567153247556704054, 6.95541710936935848460904932973, 8.568714136313185341680481310298, 9.179126293962298137971294530144, 9.782180457638577725155240553673, 10.93204797627350505314860705148
