| L(s) = 1 | + (1.01 + 0.986i)2-s + (0.422 + 1.67i)3-s + (0.0553 + 1.99i)4-s + (1.22 − 0.815i)5-s + (−1.22 + 2.11i)6-s + (0.731 − 1.76i)7-s + (−1.91 + 2.08i)8-s + (−2.64 + 1.41i)9-s + (2.04 + 0.376i)10-s + (1.20 − 6.04i)11-s + (−3.33 + 0.937i)12-s + (−3.02 + 4.53i)13-s + (2.48 − 1.06i)14-s + (1.88 + 1.70i)15-s + (−3.99 + 0.221i)16-s + (−3.03 − 3.03i)17-s + ⋯ |
| L(s) = 1 | + (0.716 + 0.697i)2-s + (0.243 + 0.969i)3-s + (0.0276 + 0.999i)4-s + (0.545 − 0.364i)5-s + (−0.501 + 0.865i)6-s + (0.276 − 0.667i)7-s + (−0.677 + 0.735i)8-s + (−0.881 + 0.472i)9-s + (0.645 + 0.119i)10-s + (0.362 − 1.82i)11-s + (−0.962 + 0.270i)12-s + (−0.839 + 1.25i)13-s + (0.663 − 0.285i)14-s + (0.486 + 0.440i)15-s + (−0.998 + 0.0552i)16-s + (−0.737 − 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0347 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0347 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28102 + 1.32636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28102 + 1.32636i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.01 - 0.986i)T \) |
| 3 | \( 1 + (-0.422 - 1.67i)T \) |
| good | 5 | \( 1 + (-1.22 + 0.815i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.731 + 1.76i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.20 + 6.04i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (3.02 - 4.53i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (3.03 + 3.03i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.32 + 3.48i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.64 - 3.98i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 0.212i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 + (-4.53 + 3.03i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.77 + 0.735i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.201 + 1.01i)T + (-39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (7.29 - 7.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.51 + 1.89i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-0.917 - 1.37i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (6.16 - 1.22i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (0.485 + 2.44i)T + (-61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-3.68 - 1.52i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.00 - 0.830i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (1.55 - 1.55i)T - 79iT^{2} \) |
| 83 | \( 1 + (12.4 + 8.32i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-2.73 - 1.13i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 8.88iT - 97T^{2} \) |
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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
−13.33423835853400723568527280911, −11.56610499381977689590505785270, −11.17971924631530019182782184609, −9.441295150084465307835116808035, −8.933392241620389004211301861354, −7.64586992097248240249788072251, −6.33920741030758664847388344124, −5.16111272995446342843330479071, −4.29693533588930181547839296610, −2.96818443864919681032593668257,
1.86922058755593424031409796849, 2.77826032644332579474623731038, 4.69298245103574087566465185556, 5.94052219626415244879045964542, 6.86923852582516347232013592529, 8.184720253454463851805208227249, 9.617470301804937379657885341751, 10.33086076013913135409926399450, 11.72531220665193004079160722347, 12.46974977665077157562763473322
