| L(s) = 1 | + (0.132 + 0.111i)2-s + (−0.342 − 1.94i)4-s + (3.51 + 1.27i)5-s + (0.526 − 2.98i)7-s + (0.343 − 0.594i)8-s + (0.323 + 0.559i)10-s + (2.34 − 0.852i)11-s + (−0.586 + 0.491i)13-s + (0.401 − 0.337i)14-s + (−3.59 + 1.30i)16-s + (−2.31 − 4.00i)17-s + (0.305 − 0.529i)19-s + (1.27 − 7.24i)20-s + (0.405 + 0.147i)22-s + (1.13 + 6.42i)23-s + ⋯ |
| L(s) = 1 | + (0.0936 + 0.0786i)2-s + (−0.171 − 0.970i)4-s + (1.57 + 0.571i)5-s + (0.198 − 1.12i)7-s + (0.121 − 0.210i)8-s + (0.102 + 0.177i)10-s + (0.705 − 0.256i)11-s + (−0.162 + 0.136i)13-s + (0.107 − 0.0900i)14-s + (−0.897 + 0.326i)16-s + (−0.560 − 0.970i)17-s + (0.0701 − 0.121i)19-s + (0.285 − 1.62i)20-s + (0.0863 + 0.0314i)22-s + (0.236 + 1.33i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82277 - 0.915432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82277 - 0.915432i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.132 - 0.111i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-3.51 - 1.27i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.526 + 2.98i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.34 + 0.852i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.586 - 0.491i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.31 + 4.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.305 + 0.529i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.13 - 6.42i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.01 + 4.21i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.13 - 6.45i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (2.47 + 4.29i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.02 + 3.38i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.23 - 1.90i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.192 + 1.09i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.84T + 53T^{2} \) |
| 59 | \( 1 + (-11.1 - 4.05i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 8.06i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.928 - 0.779i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.45 - 4.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.14 - 3.72i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.03 + 7.58i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.90 - 5.79i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (3.76 - 6.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.891 + 0.324i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.17831885378273593804696578214, −9.569305236109161772151919718508, −8.955749651130239692350469996318, −7.22149113100862865022404105980, −6.75966240772547210575990055015, −5.77428935519421604652656539682, −5.06189145569983154938761854857, −3.81412784179378410634975036813, −2.24090400932524626014469689624, −1.14168291110487845775563980064,
1.83126215040501387098833357912, 2.63242502792029261610946183191, 4.14434358868058666180212811249, 5.15368560289177459712530602054, 6.00422369799708670666529376059, 6.87170390037250764282678421788, 8.306700719361110820448637526172, 8.822666590475607975541423426227, 9.437109216759925918658804850432, 10.39382052283483421442609015390
