| L(s) = 1 | − 3.90i·2-s − 11.2·4-s − 3.88i·5-s − 9.05·7-s + 28.3i·8-s − 15.1·10-s − 8.82i·11-s − 14.4·13-s + 35.3i·14-s + 65.7·16-s − 4.12i·17-s + 28.8·19-s + 43.7i·20-s − 34.4·22-s + 0.907i·23-s + ⋯ |
| L(s) = 1 | − 1.95i·2-s − 2.81·4-s − 0.777i·5-s − 1.29·7-s + 3.54i·8-s − 1.51·10-s − 0.802i·11-s − 1.10·13-s + 2.52i·14-s + 4.11·16-s − 0.242i·17-s + 1.51·19-s + 2.18i·20-s − 1.56·22-s + 0.0394i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04546448704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04546448704\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 4.12iT \) |
| good | 2 | \( 1 + 3.90iT - 4T^{2} \) |
| 5 | \( 1 + 3.88iT - 25T^{2} \) |
| 7 | \( 1 + 9.05T + 49T^{2} \) |
| 11 | \( 1 + 8.82iT - 121T^{2} \) |
| 13 | \( 1 + 14.4T + 169T^{2} \) |
| 19 | \( 1 - 28.8T + 361T^{2} \) |
| 23 | \( 1 - 0.907iT - 529T^{2} \) |
| 29 | \( 1 - 46.7iT - 841T^{2} \) |
| 31 | \( 1 + 31.6T + 961T^{2} \) |
| 37 | \( 1 + 26.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 4.19iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 47.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 55.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 59.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 2.65iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 34.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 121.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 13.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 88.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95.4T + 9.40e3T^{2} \) |
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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
−10.86366157006101175532340817667, −10.01697983716251889715968665937, −9.252188923440204185800715629853, −8.855639577468211337938466826517, −7.42185399443482484722386400688, −5.60462316413921953276510866475, −4.80913482896492837342188889136, −3.48619425832600089085976707450, −2.83856297890201795808200241778, −1.19422206778325588380870979371,
0.02169365623033382022236071508, 3.12752152843331608052663777044, 4.35158777927892748216028210887, 5.50796138257680321363051892091, 6.34345747096727469195253619583, 7.29646303254547275868835673878, 7.47430520878702310907262742673, 8.999060496242327158356375239591, 9.700791034990064693433857382589, 10.24562011726508225271411455798
