| L(s) = 1 | + (0.853 − 1.01i)2-s + (25.2 − 9.68i)3-s + (10.8 + 61.2i)4-s + (−55.1 + 151. i)5-s + (11.6 − 33.8i)6-s + (−3.08 + 17.5i)7-s + (145. + 83.7i)8-s + (541. − 488. i)9-s + (107. + 185. i)10-s + (404. + 1.11e3i)11-s + (866. + 1.44e3i)12-s + (466. − 391. i)13-s + (15.1 + 18.0i)14-s + (77.9 + 4.35e3i)15-s + (−3.53e3 + 1.28e3i)16-s + (3.81e3 − 2.20e3i)17-s + ⋯ |
| L(s) = 1 | + (0.106 − 0.127i)2-s + (0.933 − 0.358i)3-s + (0.168 + 0.957i)4-s + (−0.441 + 1.21i)5-s + (0.0539 − 0.156i)6-s + (−0.00899 + 0.0510i)7-s + (0.283 + 0.163i)8-s + (0.742 − 0.669i)9-s + (0.107 + 0.185i)10-s + (0.304 + 0.835i)11-s + (0.501 + 0.833i)12-s + (0.212 − 0.178i)13-s + (0.00552 + 0.00658i)14-s + (0.0231 + 1.29i)15-s + (−0.862 + 0.314i)16-s + (0.775 − 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.98314 + 0.919367i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98314 + 0.919367i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-25.2 + 9.68i)T \) |
| good | 2 | \( 1 + (-0.853 + 1.01i)T + (-11.1 - 63.0i)T^{2} \) |
| 5 | \( 1 + (55.1 - 151. i)T + (-1.19e4 - 1.00e4i)T^{2} \) |
| 7 | \( 1 + (3.08 - 17.5i)T + (-1.10e5 - 4.02e4i)T^{2} \) |
| 11 | \( 1 + (-404. - 1.11e3i)T + (-1.35e6 + 1.13e6i)T^{2} \) |
| 13 | \( 1 + (-466. + 391. i)T + (8.38e5 - 4.75e6i)T^{2} \) |
| 17 | \( 1 + (-3.81e3 + 2.20e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-5.74e3 + 9.94e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (1.32e4 - 2.34e3i)T + (1.39e8 - 5.06e7i)T^{2} \) |
| 29 | \( 1 + (1.73e4 - 2.06e4i)T + (-1.03e8 - 5.85e8i)T^{2} \) |
| 31 | \( 1 + (7.22e3 + 4.09e4i)T + (-8.33e8 + 3.03e8i)T^{2} \) |
| 37 | \( 1 + (1.32e4 + 2.29e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-4.12e4 - 4.91e4i)T + (-8.24e8 + 4.67e9i)T^{2} \) |
| 43 | \( 1 + (-9.29e4 + 3.38e4i)T + (4.84e9 - 4.06e9i)T^{2} \) |
| 47 | \( 1 + (4.52e4 + 7.97e3i)T + (1.01e10 + 3.68e9i)T^{2} \) |
| 53 | \( 1 + 4.06e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-7.52e4 + 2.06e5i)T + (-3.23e10 - 2.71e10i)T^{2} \) |
| 61 | \( 1 + (-2.59e4 + 1.47e5i)T + (-4.84e10 - 1.76e10i)T^{2} \) |
| 67 | \( 1 + (4.25e5 - 3.57e5i)T + (1.57e10 - 8.90e10i)T^{2} \) |
| 71 | \( 1 + (-6.00e5 + 3.46e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (8.73e4 - 1.51e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-2.38e5 - 2.00e5i)T + (4.22e10 + 2.39e11i)T^{2} \) |
| 83 | \( 1 + (4.41e5 - 5.26e5i)T + (-5.67e10 - 3.21e11i)T^{2} \) |
| 89 | \( 1 + (-2.75e5 - 1.58e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-6.64e5 + 2.41e5i)T + (6.38e11 - 5.35e11i)T^{2} \) |
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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
−15.92317529094893610622870682844, −14.86945867310457241341325813197, −13.75081794731054125773598860530, −12.45150020322878370358002167389, −11.26435292145012161507262736448, −9.452049541475830030311607300154, −7.72777110880633967957869852103, −7.03509681949993609276289309798, −3.76912634082221410644018036290, −2.55383594766910186619557270234,
1.31252720078521996395792997550, 3.98078146634163601125024386105, 5.63270462509735330368379307722, 7.924206358628874290343302290819, 9.128965745076524655116081813683, 10.37174468775860547918118498021, 12.14266182868393906181522256437, 13.70564429499387360284423048130, 14.54761467801909523922305467173, 15.94951920620294894411624762665
