| L(s) = 1 | + (2.08 + 0.182i)2-s + (−0.507 + 1.65i)3-s + (2.35 + 0.414i)4-s + (−1.98 − 1.03i)5-s + (−1.36 + 3.36i)6-s + (3.38 + 2.36i)7-s + (0.784 + 0.210i)8-s + (−2.48 − 1.68i)9-s + (−3.95 − 2.51i)10-s + (1.66 − 4.58i)11-s + (−1.87 + 3.68i)12-s + (−0.223 − 2.55i)13-s + (6.62 + 5.55i)14-s + (2.71 − 2.76i)15-s + (−2.88 − 1.05i)16-s + (−2.36 + 0.632i)17-s + ⋯ |
| L(s) = 1 | + (1.47 + 0.129i)2-s + (−0.292 + 0.956i)3-s + (1.17 + 0.207i)4-s + (−0.887 − 0.461i)5-s + (−0.555 + 1.37i)6-s + (1.27 + 0.894i)7-s + (0.277 + 0.0743i)8-s + (−0.828 − 0.560i)9-s + (−1.24 − 0.795i)10-s + (0.503 − 1.38i)11-s + (−0.542 + 1.06i)12-s + (−0.0619 − 0.707i)13-s + (1.77 + 1.48i)14-s + (0.701 − 0.713i)15-s + (−0.722 − 0.262i)16-s + (−0.572 + 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74563 + 0.688371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74563 + 0.688371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.507 - 1.65i)T \) |
| 5 | \( 1 + (1.98 + 1.03i)T \) |
| good | 2 | \( 1 + (-2.08 - 0.182i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-3.38 - 2.36i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.66 + 4.58i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.223 + 2.55i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (2.36 - 0.632i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.05 - 2.34i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.13 - 3.04i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-2.44 + 2.05i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.272 - 1.54i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.592 - 2.21i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (4.38 - 5.22i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.553 - 1.18i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.220 - 0.314i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-0.177 + 0.177i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.797 - 0.290i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.38 - 7.86i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.09 + 0.708i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (7.83 + 4.52i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.19 + 11.9i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (6.94 + 8.27i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.325 + 3.71i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-7.58 - 13.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.25 - 0.583i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.43074141481135582183583947125, −12.17973312169243679199647637737, −11.56383735410626892690524351942, −10.89617276526553654263076186935, −8.923715150123344265620370072297, −8.220786000208264827949290188792, −6.14676280321090448489722062279, −5.23102297383225691519828433752, −4.38198259131750631013684710800, −3.23878201602278915039237959874,
2.17282611222781939328311930250, 4.17332711705869399691945804548, 4.83958155489270386944199328108, 6.73659079611046852514506668946, 7.15726392632881170844551444740, 8.532155290865778631344113131043, 10.79860835212392261760027848115, 11.44453382843662213265907879596, 12.20278360659705974348636658736, 13.05023585474172508993335983165
