| L(s) = 1 | + (−1.5 − 0.866i)3-s + (−2.31 + 1.33i)5-s + (6.66 − 2.13i)7-s + (1.5 + 2.59i)9-s + (5.85 − 10.1i)11-s − 20.4i·13-s + 4.63·15-s + (−14.2 − 8.21i)17-s + (26.9 − 15.5i)19-s + (−11.8 − 2.57i)21-s + (2.82 + 4.89i)23-s + (−8.91 + 15.4i)25-s − 5.19i·27-s + 21.6·29-s + (−4.60 − 2.65i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.288i)3-s + (−0.463 + 0.267i)5-s + (0.952 − 0.304i)7-s + (0.166 + 0.288i)9-s + (0.531 − 0.921i)11-s − 1.57i·13-s + 0.308·15-s + (−0.837 − 0.483i)17-s + (1.41 − 0.818i)19-s + (−0.564 − 0.122i)21-s + (0.122 + 0.212i)23-s + (−0.356 + 0.617i)25-s − 0.192i·27-s + 0.746·29-s + (−0.148 − 0.0857i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02834 - 0.655377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02834 - 0.655377i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (-6.66 + 2.13i)T \) |
| good | 5 | \( 1 + (2.31 - 1.33i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-5.85 + 10.1i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 20.4iT - 169T^{2} \) |
| 17 | \( 1 + (14.2 + 8.21i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-26.9 + 15.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-2.82 - 4.89i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 21.6T + 841T^{2} \) |
| 31 | \( 1 + (4.60 + 2.65i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (9.64 + 16.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 28.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 73.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + (27.9 - 16.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (47.3 - 82.0i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (97.8 + 56.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (30 - 17.3i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-3.84 + 6.65i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 22.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (5.83 + 3.37i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-23.5 - 40.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 39.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-87.5 + 50.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 30.0iT - 9.40e3T^{2} \) |
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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
−12.21067047516377159856232789586, −11.18637138379464793289978314007, −10.88933880441071401892830462468, −9.303463320279947567830246463712, −7.998747817428375022771866100050, −7.28790785716971607623447259276, −5.86721238964887469673297242504, −4.76098316946319380751964944855, −3.13608882854240365148848919335, −0.887586230714801006567079825071,
1.73233168854476431102271607833, 4.10996778585643199135857649549, 4.86642043747093376909499313203, 6.35287379940609543932633655435, 7.52090711808703030836097174866, 8.740652491437481838235265580611, 9.677829459546847999505489347455, 10.94805548161677867778112131894, 11.89027436333818256378886221530, 12.24018336203538829102337046871
