| L(s) = 1 | + (8.90 − 6.46i)2-s + (−8.34 − 25.6i)3-s + (−2.11 + 6.50i)4-s + (−328. − 239. i)5-s + (−240. − 174. i)6-s + (−385. + 1.18e3i)7-s + (458. + 1.41e3i)8-s + (−589. + 428. i)9-s − 4.47e3·10-s + (−447. − 4.39e3i)11-s + 184.·12-s + (−4.56e3 + 3.31e3i)13-s + (4.24e3 + 1.30e4i)14-s + (−3.39e3 + 1.04e4i)15-s + (1.25e4 + 9.08e3i)16-s + (−1.87e4 − 1.36e4i)17-s + ⋯ |
| L(s) = 1 | + (0.787 − 0.571i)2-s + (−0.178 − 0.549i)3-s + (−0.0165 + 0.0508i)4-s + (−1.17 − 0.855i)5-s + (−0.454 − 0.330i)6-s + (−0.425 + 1.30i)7-s + (0.316 + 0.974i)8-s + (−0.269 + 0.195i)9-s − 1.41·10-s + (−0.101 − 0.994i)11-s + 0.0308·12-s + (−0.575 + 0.418i)13-s + (0.413 + 1.27i)14-s + (−0.259 + 0.798i)15-s + (0.763 + 0.554i)16-s + (−0.924 − 0.671i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00181904 + 0.00394140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00181904 + 0.00394140i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.34 + 25.6i)T \) |
| 11 | \( 1 + (447. + 4.39e3i)T \) |
| good | 2 | \( 1 + (-8.90 + 6.46i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (328. + 239. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (385. - 1.18e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (4.56e3 - 3.31e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.87e4 + 1.36e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-3.33e3 - 1.02e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 8.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-5.96e4 + 1.83e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (9.20e4 - 6.68e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.39e5 + 4.30e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-5.85e4 - 1.80e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 1.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.49e5 - 4.60e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-5.95e5 + 4.32e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (7.74e5 - 2.38e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.01e6 + 7.34e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 2.01e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-3.92e6 - 2.85e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.72e6 - 5.31e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-5.30e6 + 3.85e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (4.63e6 + 3.36e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 8.34e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.98e6 - 3.62e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−15.68075050957779869130847426643, −14.04202067274353737886896543752, −12.80812752461096355402218941201, −12.02450289594619189156765648405, −11.46312705098295056311296387699, −8.912449315600896710288367265591, −7.899367456480614972025018388891, −5.72280635631910921710372266415, −4.21364159107256118881582830983, −2.50821724531638959050898130156,
0.00153324536923741955074693611, 3.67632080091091146171032269194, 4.58941086598351336105948405216, 6.61998249646783574192256933695, 7.54122185307319112818473968963, 9.990396202542120240155465280011, 10.79980854157587221011930567122, 12.44791410929543193232254372728, 13.84051967683781737280662205117, 14.97552567100449525094891333844
