| L(s) = 1 | + (−0.196 − 1.40i)2-s + (0.0178 + 0.181i)3-s + (−1.92 + 0.551i)4-s + (−1.42 + 2.66i)5-s + (0.250 − 0.0608i)6-s + (−1.80 − 1.93i)7-s + (1.15 + 2.58i)8-s + (2.90 − 0.578i)9-s + (4.01 + 1.47i)10-s + (2.82 − 2.31i)11-s + (−0.134 − 0.339i)12-s + (−1.74 + 0.930i)13-s + (−2.35 + 2.91i)14-s + (−0.510 − 0.211i)15-s + (3.39 − 2.12i)16-s + (−6.82 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.139 − 0.990i)2-s + (0.0103 + 0.104i)3-s + (−0.961 + 0.275i)4-s + (−0.637 + 1.19i)5-s + (0.102 − 0.0248i)6-s + (−0.682 − 0.730i)7-s + (0.407 + 0.913i)8-s + (0.969 − 0.192i)9-s + (1.27 + 0.465i)10-s + (0.850 − 0.697i)11-s + (−0.0388 − 0.0979i)12-s + (−0.482 + 0.257i)13-s + (−0.628 + 0.777i)14-s + (−0.131 − 0.0545i)15-s + (0.847 − 0.530i)16-s + (−1.65 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0257799 + 0.0868334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0257799 + 0.0868334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.196 + 1.40i)T \) |
| 7 | \( 1 + (1.80 + 1.93i)T \) |
| good | 3 | \( 1 + (-0.0178 - 0.181i)T + (-2.94 + 0.585i)T^{2} \) |
| 5 | \( 1 + (1.42 - 2.66i)T + (-2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (-2.82 + 2.31i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.74 - 0.930i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (6.82 - 2.82i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.999 + 3.29i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (1.40 - 2.10i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (7.43 + 6.10i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-3.15 - 3.15i)T + 31iT^{2} \) |
| 37 | \( 1 + (6.75 - 2.05i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (6.27 + 4.19i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (8.66 + 0.853i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (11.1 - 4.61i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.87 + 1.53i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (0.516 - 0.965i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (0.898 + 9.12i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (1.26 + 12.8i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (1.79 - 9.00i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-12.2 + 2.43i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.855 + 2.06i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.66 - 0.506i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (1.57 + 2.35i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (6.70 - 6.70i)T - 97iT^{2} \) |
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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
−9.807884521274725138144450120494, −9.096799904643765773345673990960, −7.989806970470274822188578235584, −6.93110789129469241644954150734, −6.53221083519321703808972820121, −4.70057715920135487451759882616, −3.77038775656907178731162108942, −3.34492702875129428607719438062, −1.86910355177516520357119063433, −0.04553933564960346059556997442,
1.69586509261485089203539699045, 3.76499000631903482050849791309, 4.63417183448139874457245886810, 5.25977088510768763285114604663, 6.59151799427096492225308902888, 7.08691720848247697312151263148, 8.143834575990008672546925518131, 8.859852098997260479451703690096, 9.488685208055308532543058156273, 10.16458529640541271868207676959
