| L(s) = 1 | + (−12.9 − 2.28i)2-s + (19.5 + 18.6i)3-s + (101. + 37.0i)4-s + (−78.6 − 93.6i)5-s + (−209. − 285. i)6-s + (149. − 54.5i)7-s + (−504. − 291. i)8-s + (32.9 + 728. i)9-s + (802. + 1.39e3i)10-s + (−771. + 919. i)11-s + (1.29e3 + 2.62e3i)12-s + (595. + 3.37e3i)13-s + (−2.06e3 + 363. i)14-s + (213. − 3.29e3i)15-s + (550. + 461. i)16-s + (194. − 112. i)17-s + ⋯ |
| L(s) = 1 | + (−1.61 − 0.285i)2-s + (0.722 + 0.690i)3-s + (1.59 + 0.579i)4-s + (−0.628 − 0.749i)5-s + (−0.971 − 1.32i)6-s + (0.437 − 0.159i)7-s + (−0.986 − 0.569i)8-s + (0.0451 + 0.998i)9-s + (0.802 + 1.39i)10-s + (−0.579 + 0.690i)11-s + (0.750 + 1.51i)12-s + (0.271 + 1.53i)13-s + (−0.751 + 0.132i)14-s + (0.0632 − 0.976i)15-s + (0.134 + 0.112i)16-s + (0.0395 − 0.0228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.429746 + 0.473516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429746 + 0.473516i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-19.5 - 18.6i)T \) |
| good | 2 | \( 1 + (12.9 + 2.28i)T + (60.1 + 21.8i)T^{2} \) |
| 5 | \( 1 + (78.6 + 93.6i)T + (-2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-149. + 54.5i)T + (9.01e4 - 7.56e4i)T^{2} \) |
| 11 | \( 1 + (771. - 919. i)T + (-3.07e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (-595. - 3.37e3i)T + (-4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-194. + 112. i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.50e3 - 7.81e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (4.90e3 - 1.34e4i)T + (-1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (8.34e3 + 1.47e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-5.30e4 - 1.93e4i)T + (6.79e8 + 5.70e8i)T^{2} \) |
| 37 | \( 1 + (4.41e4 + 7.64e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-9.36e4 + 1.65e4i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (5.48e4 + 4.60e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-3.18e4 - 8.73e4i)T + (-8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 - 7.62e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-4.54e4 - 5.41e4i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (-2.26e4 + 8.25e3i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (1.59e4 + 9.05e4i)T + (-8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (3.88e4 - 2.24e4i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.41e5 + 2.44e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-4.28e4 + 2.43e5i)T + (-2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (4.40e5 + 7.75e4i)T + (3.07e11 + 1.11e11i)T^{2} \) |
| 89 | \( 1 + (-7.75e5 - 4.48e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (3.90e5 + 3.27e5i)T + (1.44e11 + 8.20e11i)T^{2} \) |
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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
−16.39456147077247307916197348861, −15.71145691166522821672672835406, −14.08558428960637446429504083171, −12.07777986331790323217148620966, −10.74033330533206998726733871806, −9.548880146185517689487487611564, −8.521020683676147333076990029661, −7.59481313828490230975035574690, −4.34168239854025276650959094014, −1.82668077442783498655451492163,
0.56380209732906477717507778007, 2.74661651623420360351967889773, 6.58826640149556793055408891934, 7.940130400973297902909361602016, 8.464005807907329632715965861778, 10.25249087169162977516762212746, 11.38233603965060887418347148887, 13.22730059907003879029250137739, 14.96351680688850292831474675266, 15.65872272235003796715694060178
