| L(s) = 1 | + (0.686 − 1.23i)2-s + (−1.64 − 0.537i)3-s + (−1.05 − 1.69i)4-s + (1.64 + 1.08i)5-s + (−1.79 + 1.66i)6-s + (3.20 − 3.02i)7-s + (−2.82 + 0.142i)8-s + (2.42 + 1.76i)9-s + (2.46 − 1.29i)10-s + (0.555 − 1.10i)11-s + (0.830 + 3.36i)12-s + (5.40 + 2.32i)13-s + (−1.53 − 6.03i)14-s + (−2.12 − 2.66i)15-s + (−1.76 + 3.59i)16-s + (−0.839 + 1.00i)17-s + ⋯ |
| L(s) = 1 | + (0.485 − 0.874i)2-s + (−0.950 − 0.310i)3-s + (−0.528 − 0.848i)4-s + (0.736 + 0.484i)5-s + (−0.732 + 0.680i)6-s + (1.21 − 1.14i)7-s + (−0.998 + 0.0504i)8-s + (0.807 + 0.589i)9-s + (0.780 − 0.408i)10-s + (0.167 − 0.333i)11-s + (0.239 + 0.970i)12-s + (1.49 + 0.646i)13-s + (−0.411 − 1.61i)14-s + (−0.549 − 0.688i)15-s + (−0.440 + 0.897i)16-s + (−0.203 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.944789 - 1.48082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.944789 - 1.48082i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.686 + 1.23i)T \) |
| 3 | \( 1 + (1.64 + 0.537i)T \) |
| good | 5 | \( 1 + (-1.64 - 1.08i)T + (1.98 + 4.59i)T^{2} \) |
| 7 | \( 1 + (-3.20 + 3.02i)T + (0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.555 + 1.10i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-5.40 - 2.32i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (0.839 - 1.00i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.869 + 0.729i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.24 + 2.38i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (9.66 + 1.12i)T + (28.2 + 6.68i)T^{2} \) |
| 31 | \( 1 + (-0.734 + 3.09i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (4.78 + 0.843i)T + (34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-8.90 + 6.63i)T + (11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.456 - 7.82i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (9.28 - 2.20i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (3.53 - 6.11i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.88 - 5.74i)T + (-35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-7.14 - 2.13i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (-6.40 + 0.748i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (-1.77 + 0.646i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.89 + 2.87i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (5.76 + 4.29i)T + (22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (7.58 + 5.64i)T + (23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (4.69 - 12.9i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.18 + 6.04i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.74451947553635687733356100679, −9.839055828454065873018940016298, −8.702152945207710218920438026648, −7.44877078637712271856589499797, −6.37005592979472960620605693450, −5.73629075698388342897575344077, −4.58730363108688642072893774261, −3.82937418254794547887674462176, −1.99047919453574613464998753409, −1.08925313066468543115343805339,
1.62331148288848681920510898240, 3.59895958086704455392180859050, 4.87693287465153530944928507321, 5.48112239604662335221585187962, 5.92289694792713650408442418465, 7.09270836436932762075662713944, 8.260858509638953208109538758913, 8.965566119305541812001365996644, 9.757573865680990824477811646901, 11.23412629759110557096612057795
