| L(s) = 1 | + (1.13 + 0.845i)2-s + (−1.46 + 0.917i)3-s + (0.569 + 1.91i)4-s + (0.908 − 1.35i)5-s + (−2.44 − 0.201i)6-s + (1.44 + 3.49i)7-s + (−0.975 + 2.65i)8-s + (1.31 − 2.69i)9-s + (2.17 − 0.772i)10-s + (−3.39 + 0.675i)11-s + (−2.59 − 2.29i)12-s + (−1.72 + 1.15i)13-s + (−1.31 + 5.18i)14-s + (−0.0863 + 2.82i)15-s + (−3.35 + 2.18i)16-s + (1.04 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (0.801 + 0.597i)2-s + (−0.848 + 0.529i)3-s + (0.284 + 0.958i)4-s + (0.406 − 0.607i)5-s + (−0.996 − 0.0822i)6-s + (0.547 + 1.32i)7-s + (−0.344 + 0.938i)8-s + (0.438 − 0.898i)9-s + (0.688 − 0.244i)10-s + (−1.02 + 0.203i)11-s + (−0.749 − 0.661i)12-s + (−0.477 + 0.319i)13-s + (−0.351 + 1.38i)14-s + (−0.0222 + 0.730i)15-s + (−0.837 + 0.546i)16-s + (0.254 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.965425 + 1.12138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.965425 + 1.12138i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.13 - 0.845i)T \) |
| 3 | \( 1 + (1.46 - 0.917i)T \) |
| good | 5 | \( 1 + (-0.908 + 1.35i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.44 - 3.49i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (3.39 - 0.675i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.72 - 1.15i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-1.04 + 1.04i)T - 17iT^{2} \) |
| 19 | \( 1 + (-6.92 + 4.62i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 4.54i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.76 + 8.89i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + (-1.65 + 2.47i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.72 - 0.713i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (9.26 - 1.84i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (0.0451 + 0.0451i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.12 - 5.67i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (5.63 + 3.76i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-1.08 + 5.44i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (8.44 + 1.67i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (7.65 - 3.16i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-6.47 - 2.68i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (4.09 + 4.09i)T + 79iT^{2} \) |
| 83 | \( 1 + (-3.39 - 5.07i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (2.79 - 1.15i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 5.24iT - 97T^{2} \) |
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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
−12.70948856685989585555314184742, −11.93756605573543581788737760932, −11.28180301798989764511638595711, −9.737139858756364787851398017113, −8.803856203254561130091451508270, −7.52995392719229922757932182310, −6.14135657766646522089303557248, −5.13468028257610572476725286018, −4.82270784577291699548531710030, −2.70332771322452906523422384065,
1.35572446044170494206224679652, 3.18451307629382484344693390047, 4.85325432647525835555800498018, 5.67523793582557770054559514130, 6.97505218396472192640614199223, 7.72831607189940292442212175716, 10.15674431624026908456436165398, 10.37600777194497827140052261459, 11.33556870351336754272359494092, 12.23962673175445008592857339789
