| L(s) = 1 | + (−3.24 − 0.571i)2-s + (−71.7 + 37.5i)3-s + (−230. − 83.8i)4-s + (387. + 462. i)5-s + (254. − 80.8i)6-s + (−3.86e3 + 1.40e3i)7-s + (1.42e3 + 824. i)8-s + (3.73e3 − 5.39e3i)9-s + (−992. − 1.71e3i)10-s + (1.14e4 − 1.36e4i)11-s + (1.96e4 − 2.64e3i)12-s + (−882. − 5.00e3i)13-s + (1.33e4 − 2.34e3i)14-s + (−4.51e4 − 1.85e4i)15-s + (4.39e4 + 3.68e4i)16-s + (2.37e4 − 1.37e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.202 − 0.0357i)2-s + (−0.885 + 0.464i)3-s + (−0.899 − 0.327i)4-s + (0.620 + 0.739i)5-s + (0.196 − 0.0624i)6-s + (−1.60 + 0.585i)7-s + (0.348 + 0.201i)8-s + (0.569 − 0.822i)9-s + (−0.0992 − 0.171i)10-s + (0.780 − 0.929i)11-s + (0.949 − 0.127i)12-s + (−0.0309 − 0.175i)13-s + (0.346 − 0.0611i)14-s + (−0.892 − 0.366i)15-s + (0.670 + 0.562i)16-s + (0.284 − 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.568334 - 0.267235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568334 - 0.267235i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (71.7 - 37.5i)T \) |
| good | 2 | \( 1 + (3.24 + 0.571i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (-387. - 462. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (3.86e3 - 1.40e3i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (-1.14e4 + 1.36e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (882. + 5.00e3i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (-2.37e4 + 1.37e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.28e4 + 1.60e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (8.84e4 - 2.42e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (2.34e5 + 4.13e4i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (6.18e5 + 2.25e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (-7.79e5 - 1.34e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-3.82e6 + 6.74e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (1.19e6 + 1.00e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (-4.07e5 - 1.11e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 + 1.29e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.38e7 + 1.64e7i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-1.39e7 + 5.08e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (1.06e6 + 6.02e6i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (-3.07e7 + 1.77e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-6.89e6 + 1.19e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (7.76e6 - 4.40e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (4.57e7 + 8.06e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (3.60e6 + 2.07e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (1.08e8 + 9.11e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−15.50289245434849185163778339978, −14.02753898811541058174037183323, −12.86027015672748070501124119823, −11.24925801115096068453691638471, −9.828839237362547277343521098485, −9.295111760419860545930808587346, −6.51318000328281952650019164453, −5.55164827094441464182279222026, −3.43620416092634031472591172821, −0.44284338194401356182513303091,
1.08563926545434184245788875173, 4.14304162418407347692074149470, 5.84573925981382798584668601895, 7.32607269686129666405258099781, 9.323273402819411225744729892507, 10.11749896746970166148046855748, 12.41382615477802163707278764199, 12.83889061683609257772542749077, 14.02774465241804346926794518106, 16.48527810273319573479835653562
