| L(s) = 1 | + (1.30 − 1.13i)3-s + (−0.759 + 2.08i)5-s + (1.09 + 0.192i)7-s + (0.420 − 2.97i)9-s + (1.84 − 0.670i)11-s + (−2.48 + 2.08i)13-s + (1.37 + 3.59i)15-s + (3.10 − 1.79i)17-s + (6.22 + 3.59i)19-s + (1.65 − 0.990i)21-s + (−0.119 − 0.676i)23-s + (0.0543 + 0.0455i)25-s + (−2.82 − 4.36i)27-s + (5.21 − 6.21i)29-s + (2.39 − 0.422i)31-s + ⋯ |
| L(s) = 1 | + (0.755 − 0.655i)3-s + (−0.339 + 0.933i)5-s + (0.413 + 0.0729i)7-s + (0.140 − 0.990i)9-s + (0.555 − 0.202i)11-s + (−0.688 + 0.577i)13-s + (0.355 + 0.927i)15-s + (0.752 − 0.434i)17-s + (1.42 + 0.824i)19-s + (0.360 − 0.216i)21-s + (−0.0248 − 0.141i)23-s + (0.0108 + 0.00911i)25-s + (−0.543 − 0.839i)27-s + (0.968 − 1.15i)29-s + (0.430 − 0.0758i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09556 - 0.0991223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09556 - 0.0991223i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 + 1.13i)T \) |
| good | 5 | \( 1 + (0.759 - 2.08i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.09 - 0.192i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.84 + 0.670i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.48 - 2.08i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.10 + 1.79i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.22 - 3.59i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.119 + 0.676i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.21 + 6.21i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.39 + 0.422i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.40 - 5.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.84 + 5.77i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.11 - 8.55i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.60 - 9.08i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.398iT - 53T^{2} \) |
| 59 | \( 1 + (6.64 + 2.41i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.285 + 1.62i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.00 - 1.19i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.82 + 3.16i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.80 + 10.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.03 + 3.61i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.03 - 5.06i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (7.16 + 4.13i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.6 - 4.23i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.865847773178312857703944430583, −9.456345108469105724486018707021, −8.155853505144316210854158681413, −7.69864090843974336483480558409, −6.86092576617704583876177334827, −6.08853218808944455468783610813, −4.66686822394314541084944518959, −3.43769266241773712709221027930, −2.71754648911704409253645879993, −1.32193113533595849674439180741,
1.22064544882451728017057828849, 2.80030636072484971491046457580, 3.83951286756590781390332405191, 4.87091283595766381919552603956, 5.32491238446621920917300361369, 7.01042764909497830840160659653, 7.87357279122519789989017051949, 8.519279316284526509855772582500, 9.312699219338746566397745285801, 9.984014694250392205993292505996
