| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0943 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0943 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.136167 - 0.149688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.136167 - 0.149688i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.17802610581214694615980936308, −9.368744365815966208675194329693, −8.265510058974873148163182468671, −6.96174566950602171080150715621, −6.71487029906276922589181565438, −5.47096793722777788506855078344, −4.57707527266256962239545712668, −3.57636891554578996255122736386, −2.48775056871368357133072885592, −0.11308943441110161621340401736,
1.30357403879228195833930214582, 2.80431731225830757064745115937, 4.32625851365747964978872927942, 5.19632099278163046868519351428, 5.98531099671959644866497503850, 6.89602459957999870045856817914, 8.077842695460225181842213003346, 8.291017126173040359755235936838, 9.675430198017769215539297817429, 10.47444644810276188491173993911
