| L(s) = 1 | + (19.8 − 34.4i)2-s + (−534. − 925. i)4-s + (−423. − 733. i)5-s + (3.52e3 − 6.10e3i)7-s − 2.21e4·8-s − 3.36e4·10-s + (−4.50e4 + 7.80e4i)11-s + (−3.42e4 − 5.93e4i)13-s + (−1.40e5 − 2.42e5i)14-s + (−1.66e5 + 2.88e5i)16-s + 2.35e5·17-s − 2.32e5·19-s + (−4.52e5 + 7.83e5i)20-s + (1.79e6 + 3.10e6i)22-s + (−1.33e5 − 2.30e5i)23-s + ⋯ |
| L(s) = 1 | + (0.878 − 1.52i)2-s + (−1.04 − 1.80i)4-s + (−0.302 − 0.524i)5-s + (0.554 − 0.960i)7-s − 1.91·8-s − 1.06·10-s + (−0.927 + 1.60i)11-s + (−0.333 − 0.576i)13-s + (−0.974 − 1.68i)14-s + (−0.634 + 1.09i)16-s + 0.684·17-s − 0.410·19-s + (−0.632 + 1.09i)20-s + (1.63 + 2.82i)22-s + (−0.0991 − 0.171i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.618122 + 2.00890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.618122 + 2.00890i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-19.8 + 34.4i)T + (-256 - 443. i)T^{2} \) |
| 5 | \( 1 + (423. + 733. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-3.52e3 + 6.10e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.50e4 - 7.80e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (3.42e4 + 5.93e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 - 2.35e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.32e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (1.33e5 + 2.30e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-1.53e6 + 2.65e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + (1.79e6 + 3.11e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 5.13e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.25e6 + 3.89e6i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.64e7 + 2.85e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-6.21e6 + 1.07e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + 3.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (2.27e7 + 3.94e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.08e7 - 5.33e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (2.92e6 + 5.06e6i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.53e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.44e8 - 2.50e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (2.26e8 - 3.92e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 - 5.93e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-1.49e8 + 2.58e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−14.12585058042354128008209926909, −12.87853311975753766969560619563, −12.18992226263073157987734683557, −10.71952577280110265858634113208, −9.899387755808831918337634900010, −7.72283730937880260156791796510, −5.05848594622856835514537886201, −4.13616946574229129997530827974, −2.25160222315048225002218129395, −0.66339547144627913308560476100,
3.16993302636041497619025270364, 5.05683610058195686712218766501, 6.13591381436553395343405200342, 7.66686847434013068278725716483, 8.698588551907869665130045991535, 11.15339609917140101475795659408, 12.61830132921133568038158455535, 13.97828242088550587276605978684, 14.78254777202029503706250649048, 15.80797553288447547593186918493
