| L(s) = 1 | + (−18.2 + 3.22i)2-s + (61.0 − 53.2i)3-s + (83.2 − 30.2i)4-s + (−223. + 266. i)5-s + (−944. + 1.16e3i)6-s + (−1.43e3 − 523. i)7-s + (2.69e3 − 1.55e3i)8-s + (892. − 6.50e3i)9-s + (3.22e3 − 5.58e3i)10-s + (262. + 312. i)11-s + (3.46e3 − 6.27e3i)12-s + (−7.24e3 + 4.10e4i)13-s + (2.79e4 + 4.93e3i)14-s + (536. + 2.81e4i)15-s + (−6.15e4 + 5.16e4i)16-s + (5.26e4 + 3.03e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 + 0.201i)2-s + (0.753 − 0.657i)3-s + (0.325 − 0.118i)4-s + (−0.357 + 0.425i)5-s + (−0.728 + 0.902i)6-s + (−0.599 − 0.218i)7-s + (0.657 − 0.379i)8-s + (0.136 − 0.990i)9-s + (0.322 − 0.558i)10-s + (0.0179 + 0.0213i)11-s + (0.167 − 0.302i)12-s + (−0.253 + 1.43i)13-s + (0.728 + 0.128i)14-s + (0.0105 + 0.555i)15-s + (−0.939 + 0.788i)16-s + (0.630 + 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.619553 + 0.489368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619553 + 0.489368i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-61.0 + 53.2i)T \) |
| good | 2 | \( 1 + (18.2 - 3.22i)T + (240. - 87.5i)T^{2} \) |
| 5 | \( 1 + (223. - 266. i)T + (-6.78e4 - 3.84e5i)T^{2} \) |
| 7 | \( 1 + (1.43e3 + 523. i)T + (4.41e6 + 3.70e6i)T^{2} \) |
| 11 | \( 1 + (-262. - 312. i)T + (-3.72e7 + 2.11e8i)T^{2} \) |
| 13 | \( 1 + (7.24e3 - 4.10e4i)T + (-7.66e8 - 2.78e8i)T^{2} \) |
| 17 | \( 1 + (-5.26e4 - 3.03e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-8.16e4 - 1.41e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-8.85e4 - 2.43e5i)T + (-5.99e10 + 5.03e10i)T^{2} \) |
| 29 | \( 1 + (-8.33e5 + 1.46e5i)T + (4.70e11 - 1.71e11i)T^{2} \) |
| 31 | \( 1 + (7.33e5 - 2.67e5i)T + (6.53e11 - 5.48e11i)T^{2} \) |
| 37 | \( 1 + (8.58e5 - 1.48e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (2.46e4 + 4.34e3i)T + (7.50e12 + 2.73e12i)T^{2} \) |
| 43 | \( 1 + (-7.38e5 + 6.20e5i)T + (2.02e12 - 1.15e13i)T^{2} \) |
| 47 | \( 1 + (-4.63e5 + 1.27e6i)T + (-1.82e13 - 1.53e13i)T^{2} \) |
| 53 | \( 1 + 6.06e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.23e7 - 1.47e7i)T + (-2.54e13 - 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-7.35e6 - 2.67e6i)T + (1.46e14 + 1.23e14i)T^{2} \) |
| 67 | \( 1 + (3.57e6 - 2.02e7i)T + (-3.81e14 - 1.38e14i)T^{2} \) |
| 71 | \( 1 + (-1.86e7 - 1.07e7i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (2.63e7 + 4.56e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-6.77e6 - 3.84e7i)T + (-1.42e15 + 5.18e14i)T^{2} \) |
| 83 | \( 1 + (1.82e7 - 3.21e6i)T + (2.11e15 - 7.70e14i)T^{2} \) |
| 89 | \( 1 + (-7.85e7 + 4.53e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-5.90e7 + 4.95e7i)T + (1.36e15 - 7.71e15i)T^{2} \) |
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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
−16.03073895276789547964035729484, −14.51395114534015549013126226862, −13.40081070038760611228580469390, −11.90884609970989061058536038093, −10.05613838996262818139243404915, −9.029756662758913610792303402339, −7.70073727344038820573585169385, −6.78037793314255655315033887710, −3.60679268428417583267066141124, −1.42617667580181438373211399888,
0.52423694901844097185554919026, 2.86706314203625056611075044491, 4.91507936565704012872869653609, 7.66626598811386401208354779249, 8.740745598573533058024494489545, 9.741039423726593352941374141131, 10.77786276154506216606019705381, 12.67787505004297254557723277721, 14.12322621163610804571471772088, 15.60488938160402091521842760793
