| 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
−10.39382052283483421442609015390, −9.437109216759925918658804850432, −8.822666590475607975541423426227, −8.306700719361110820448637526172, −6.87170390037250764282678421788, −6.00422369799708670666529376059, −5.15368560289177459712530602054, −4.14434358868058666180212811249, −2.63242502792029261610946183191, −1.83126215040501387098833357912,
1.14168291110487845775563980064, 2.24090400932524626014469689624, 3.81412784179378410634975036813, 5.06189145569983154938761854857, 5.77428935519421604652656539682, 6.75966240772547210575990055015, 7.22149113100862865022404105980, 8.955749651130239692350469996318, 9.569305236109161772151919718508, 10.17831885378273593804696578214
