| L(s) = 1 | + (0.900 − 2.08i)2-s + (−2.17 − 2.30i)4-s + (0.0341 + 0.585i)5-s + (3.70 + 0.878i)7-s + (−2.50 + 0.912i)8-s + (1.25 + 0.456i)10-s + (4.16 + 2.74i)11-s + (2.46 + 0.288i)13-s + (5.17 − 6.94i)14-s + (0.0163 − 0.279i)16-s + (−2.18 + 1.83i)17-s + (0.319 + 0.268i)19-s + (1.27 − 1.35i)20-s + (9.48 − 6.23i)22-s + (−7.15 + 1.69i)23-s + ⋯ |
| L(s) = 1 | + (0.636 − 1.47i)2-s + (−1.08 − 1.15i)4-s + (0.0152 + 0.262i)5-s + (1.40 + 0.332i)7-s + (−0.885 + 0.322i)8-s + (0.396 + 0.144i)10-s + (1.25 + 0.826i)11-s + (0.684 + 0.0800i)13-s + (1.38 − 1.85i)14-s + (0.00407 − 0.0699i)16-s + (−0.530 + 0.445i)17-s + (0.0733 + 0.0615i)19-s + (0.285 − 0.302i)20-s + (2.02 − 1.32i)22-s + (−1.49 + 0.353i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0888 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0888 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69189 - 1.84950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69189 - 1.84950i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.900 + 2.08i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.0341 - 0.585i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-3.70 - 0.878i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (-4.16 - 2.74i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-2.46 - 0.288i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (2.18 - 1.83i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.319 - 0.268i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (7.15 - 1.69i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (3.15 + 4.23i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (0.842 - 2.81i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (1.09 + 6.19i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (1.44 + 3.35i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (5.37 - 2.70i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (1.88 + 6.30i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 2.14i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.87 - 2.54i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (7.81 - 8.28i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.55 + 4.77i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (-2.65 - 0.966i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.9 + 4.69i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.97 + 11.5i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-0.831 + 1.92i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (0.324 - 0.118i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.637 - 10.9i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.56757130720644177468163943790, −9.533199907115021057562065373737, −8.738647845115392727485844251004, −7.67348125278466058050143673781, −6.44325824366738518312587280688, −5.27492853374219141651793230134, −4.32399933311036232349224465023, −3.69820459846137366870265195505, −2.12823099080065631073350866139, −1.53035615860533331113318683229,
1.47629763714481250323365433187, 3.65461950293423397712129182168, 4.50069287151471988512452037180, 5.27143247155438961808646402696, 6.26886930410660806534482824422, 6.92198159920695296219421959900, 8.124168845213666190799193787785, 8.376095109235072495501400135688, 9.365860096970603256751137638217, 10.87673687606070763937127208801
