| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−2.10 + 1.35i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.355 − 2.47i)10-s + (1.52 + 3.33i)11-s + (0.415 + 0.909i)12-s + (−0.642 − 4.47i)13-s + (0.841 + 0.540i)14-s + (2.39 − 0.703i)15-s + (−0.142 + 0.989i)16-s + (−3.64 + 4.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.939 + 0.603i)5-s + (0.267 − 0.308i)6-s + (0.0537 − 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.112 − 0.781i)10-s + (0.458 + 1.00i)11-s + (0.119 + 0.262i)12-s + (−0.178 − 1.23i)13-s + (0.224 + 0.144i)14-s + (0.618 − 0.181i)15-s + (−0.0355 + 0.247i)16-s + (−0.884 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.680723 - 0.0680559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.680723 - 0.0680559i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.18 + 2.34i)T \) |
| good | 5 | \( 1 + (2.10 - 1.35i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-1.52 - 3.33i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.642 + 4.47i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (3.64 - 4.20i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.56 + 4.11i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-5.41 + 6.24i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.22 + 0.652i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-6.71 - 4.31i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-9.04 + 5.81i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-7.10 - 2.08i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 9.91T + 47T^{2} \) |
| 53 | \( 1 + (0.908 - 6.31i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.34 + 9.32i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.21 + 1.82i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.681 + 1.49i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.73 + 8.16i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (8.04 + 9.28i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.01 + 7.04i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-3.61 - 2.32i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (9.74 + 2.86i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-6.31 + 4.06i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.19099764676505634357913990977, −9.056132027699092411299245088995, −7.943026452964202911718561338009, −7.57200437256497392651576558012, −6.58282984648542606357015456706, −6.04611546188364967389384935231, −4.55269244456682346400143050336, −4.11694796535382788094070323810, −2.41582204091220387299870374308, −0.50824114198537327887495392093,
0.951256474162014317833830953234, 2.49919913767317610537391110283, 4.05061021549027169152870601622, 4.34763261115728291961832167364, 5.65069416141578645844297868538, 6.64043543043154025463765163495, 7.71091601509380225992357066149, 8.678765313927112474400383984519, 9.083113805361252874449545576571, 10.09142694628689021488780460967
