| L(s) = 1 | − 1.79·3-s − 0.791i·5-s + 2·7-s + 0.208·9-s + 0.791·11-s + 3.79i·13-s + 1.41i·15-s − 7.58i·17-s + 1.58i·19-s − 3.58·21-s − 3.79i·23-s + 4.37·25-s + 5.00·27-s − 3.79i·29-s − 8.37i·31-s + ⋯ |
| L(s) = 1 | − 1.03·3-s − 0.353i·5-s + 0.755·7-s + 0.0695·9-s + 0.238·11-s + 1.05i·13-s + 0.365i·15-s − 1.83i·17-s + 0.363i·19-s − 0.781·21-s − 0.790i·23-s + 0.874·25-s + 0.962·27-s − 0.704i·29-s − 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.939957 - 0.427207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939957 - 0.427207i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (-4 - 4.58i)T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 + 0.791iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 - 3.79iT - 13T^{2} \) |
| 17 | \( 1 + 7.58iT - 17T^{2} \) |
| 19 | \( 1 - 1.58iT - 19T^{2} \) |
| 23 | \( 1 + 3.79iT - 23T^{2} \) |
| 29 | \( 1 + 3.79iT - 29T^{2} \) |
| 31 | \( 1 + 8.37iT - 31T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 - 1.58iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 8.20iT - 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−10.85962911713287069133927181071, −9.668340698321600647608908970150, −8.922082224149871618247029311734, −7.85134780043768055442711614826, −6.84448360458129783732218259902, −5.94766268691423586089791474573, −4.95075261574708576097837922214, −4.32471856410510024935732975817, −2.46539414321112339234688063587, −0.77932962519536327712119101436,
1.27576840175373456860848822825, 3.02595646205313579998329251238, 4.39425633810027543452821552120, 5.45625857939269592400365381786, 6.07167231337160334549740668741, 7.14187904246922594219118915592, 8.134723461581424157321217000397, 8.975885394136751169263507164359, 10.41171213044430768694687981787, 10.78308682589471888555360519641
