| L(s) = 1 | + (−1.39 − 0.208i)2-s + (−0.339 + 1.69i)3-s + (1.91 + 0.584i)4-s + (−0.321 − 0.171i)5-s + (0.829 − 2.30i)6-s + (−1.36 − 0.270i)7-s + (−2.55 − 1.21i)8-s + (−2.76 − 1.15i)9-s + (0.413 + 0.307i)10-s + (−0.181 + 0.148i)11-s + (−1.64 + 3.05i)12-s + (−2.85 + 1.52i)13-s + (1.84 + 0.663i)14-s + (0.400 − 0.487i)15-s + (3.31 + 2.23i)16-s + (−5.78 + 2.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 − 0.147i)2-s + (−0.195 + 0.980i)3-s + (0.956 + 0.292i)4-s + (−0.143 − 0.0768i)5-s + (0.338 − 0.940i)6-s + (−0.514 − 0.102i)7-s + (−0.902 − 0.430i)8-s + (−0.923 − 0.384i)9-s + (0.130 + 0.0972i)10-s + (−0.0545 + 0.0447i)11-s + (−0.473 + 0.880i)12-s + (−0.790 + 0.422i)13-s + (0.493 + 0.177i)14-s + (0.103 − 0.125i)15-s + (0.829 + 0.559i)16-s + (−1.40 + 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00520484 - 0.0174102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00520484 - 0.0174102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.208i)T \) |
| 3 | \( 1 + (0.339 - 1.69i)T \) |
| good | 5 | \( 1 + (0.321 + 0.171i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (1.36 + 0.270i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.181 - 0.148i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (2.85 - 1.52i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (5.78 - 2.39i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.97 + 6.51i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-4.20 - 2.81i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (6.21 + 5.10i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (4.89 - 4.89i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.879 - 2.89i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (0.650 + 0.434i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.655 + 6.65i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (3.86 - 1.60i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.87 - 3.17i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-0.638 - 0.341i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.446 + 4.52i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (3.71 - 0.365i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-14.2 - 2.83i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.725 - 3.64i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (3.36 - 8.11i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (2.76 - 9.12i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (5.05 + 7.55i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-8.99 - 8.99i)T + 97iT^{2} \) |
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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
−11.36121923485795854614300629071, −10.99903260128941722728793898636, −9.842689571192762874427375753643, −9.324893135642037108460653058076, −8.547344306374314006833858525852, −7.23746179051419837967234160443, −6.38771759478938998886641204161, −5.01291902019331872156971888175, −3.74035873569420200595432559420, −2.44940732532270798718481239792,
0.01556982578353735599175063209, 1.86406952553469102310966642749, 3.12212276478749320260942144536, 5.33076210341590753264926048817, 6.29821801348233438623465652973, 7.23556172165194693403257814071, 7.80585639559693752921180962609, 8.946113386435679555558672249820, 9.706302881307998443356435227044, 10.92714868071357573110450919715
