| L(s) = 1 | + 0.480i·5-s − 5.76·11-s + 1.41i·13-s + 23.0i·17-s − 10.6i·19-s + 6.59·23-s + 24.7·25-s + 6.20·29-s − 41.9i·31-s − 60.0·37-s + 48.8i·41-s − 51.5·43-s − 19.2i·47-s − 82.1·53-s − 2.77i·55-s + ⋯ |
| L(s) = 1 | + 0.0961i·5-s − 0.523·11-s + 0.109i·13-s + 1.35i·17-s − 0.561i·19-s + 0.286·23-s + 0.990·25-s + 0.213·29-s − 1.35i·31-s − 1.62·37-s + 1.19i·41-s − 1.19·43-s − 0.409i·47-s − 1.54·53-s − 0.0503i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5220899034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5220899034\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.480iT - 25T^{2} \) |
| 11 | \( 1 + 5.76T + 121T^{2} \) |
| 13 | \( 1 - 1.41iT - 169T^{2} \) |
| 17 | \( 1 - 23.0iT - 289T^{2} \) |
| 19 | \( 1 + 10.6iT - 361T^{2} \) |
| 23 | \( 1 - 6.59T + 529T^{2} \) |
| 29 | \( 1 - 6.20T + 841T^{2} \) |
| 31 | \( 1 + 41.9iT - 961T^{2} \) |
| 37 | \( 1 + 60.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 48.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 82.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 92.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.99iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 2.20T + 4.48e3T^{2} \) |
| 71 | \( 1 - 80.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 13.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 64.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.7iT - 9.40e3T^{2} \) |
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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
−9.456216413877911537904054705257, −8.560840478231735656276459247890, −8.008828735216311609398316828739, −7.00492416502444861868649908772, −6.33188727353378693080068662811, −5.37688485970136370385785275235, −4.55403121342674087381684375746, −3.53821155376071206345475618581, −2.56351636802250735676394232345, −1.39202700864390260248649100195,
0.13583176925882985305451382070, 1.50573747664354495660925367222, 2.78028524383598079650894070356, 3.55863235201346440888870229554, 4.95922761478157750378225889154, 5.20788230574580280473538517930, 6.52671851117922250352100900886, 7.12471372086530730122555375072, 8.031013313463176141220395871633, 8.771808684268904808259813261140
