| L(s) = 1 | + (−1.36 − 0.361i)2-s + (0.881 + 0.471i)3-s + (1.73 + 0.989i)4-s + (−2.41 + 2.94i)5-s + (−1.03 − 0.963i)6-s + (−0.0684 − 0.0457i)7-s + (−2.01 − 1.98i)8-s + (0.555 + 0.831i)9-s + (4.37 − 3.15i)10-s + (0.433 + 0.131i)11-s + (1.06 + 1.69i)12-s + (−2.71 + 2.22i)13-s + (0.0769 + 0.0872i)14-s + (−3.52 + 1.45i)15-s + (2.04 + 3.43i)16-s + (−6.00 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.966 − 0.255i)2-s + (0.509 + 0.272i)3-s + (0.868 + 0.494i)4-s + (−1.08 + 1.31i)5-s + (−0.422 − 0.393i)6-s + (−0.0258 − 0.0172i)7-s + (−0.713 − 0.700i)8-s + (0.185 + 0.277i)9-s + (1.38 − 0.997i)10-s + (0.130 + 0.0396i)11-s + (0.307 + 0.488i)12-s + (−0.752 + 0.617i)13-s + (0.0205 + 0.0233i)14-s + (−0.909 + 0.376i)15-s + (0.510 + 0.859i)16-s + (−1.45 − 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.138330 + 0.459755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.138330 + 0.459755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.361i)T \) |
| 3 | \( 1 + (-0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (2.41 - 2.94i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (0.0684 + 0.0457i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.433 - 0.131i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (2.71 - 2.22i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (6.00 + 2.48i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.162 + 1.65i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (1.77 + 0.353i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.51 - 5.00i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (3.65 - 3.65i)T - 31iT^{2} \) |
| 37 | \( 1 + (-4.62 + 0.455i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (1.27 - 6.38i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (3.58 - 1.91i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (3.11 - 7.52i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.05 + 3.48i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 8.67i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (3.48 - 6.52i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (7.45 - 13.9i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-4.59 + 6.87i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-12.1 + 8.10i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.30 - 5.56i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-13.3 - 1.31i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.29 + 1.05i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (4.02 - 4.02i)T - 97iT^{2} \) |
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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
−11.37788375500131795219107405360, −10.84032462973289734858143717649, −9.879097296635171861113608191701, −8.994921867727169544528056385062, −8.051285959063524539731436667263, −7.09239111027128160832683497323, −6.68599901557694680931269505964, −4.44936477928721611530045356296, −3.28043916263963767340516798005, −2.35945900088645533057799038745,
0.38622625893522191300278138872, 2.07794240521296366061902538178, 3.85312728074426805905884801980, 5.10961965780414660731054414585, 6.47412162638839353253383078761, 7.69565350186894454857024023495, 8.143244063742243186174333663815, 8.942527528615376835177540468723, 9.719070641937497839356603419883, 10.93789454695264418212166771640
