| L(s) = 1 | + (−1.76 + 1.16i)2-s + (0.980 − 2.27i)4-s + (2.67 − 2.83i)5-s + (3.31 + 0.387i)7-s + (0.175 + 0.992i)8-s + (−1.43 + 8.11i)10-s + (0.217 − 0.724i)11-s + (−0.144 + 2.48i)13-s + (−6.30 + 3.16i)14-s + (1.93 + 2.05i)16-s + (0.700 − 0.255i)17-s + (4.21 + 1.53i)19-s + (−3.81 − 8.85i)20-s + (0.459 + 1.53i)22-s + (−2.27 + 0.265i)23-s + ⋯ |
| L(s) = 1 | + (−1.24 + 0.822i)2-s + (0.490 − 1.13i)4-s + (1.19 − 1.26i)5-s + (1.25 + 0.146i)7-s + (0.0618 + 0.350i)8-s + (−0.452 + 2.56i)10-s + (0.0654 − 0.218i)11-s + (−0.0400 + 0.688i)13-s + (−1.68 + 0.846i)14-s + (0.483 + 0.512i)16-s + (0.170 − 0.0618i)17-s + (0.967 + 0.352i)19-s + (−0.853 − 1.97i)20-s + (0.0978 + 0.327i)22-s + (−0.473 + 0.0553i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14792 + 0.0807817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14792 + 0.0807817i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.76 - 1.16i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (-2.67 + 2.83i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.31 - 0.387i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.217 + 0.724i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (0.144 - 2.48i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-0.700 + 0.255i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.21 - 1.53i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (2.27 - 0.265i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (0.414 + 0.208i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (2.35 + 3.16i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (3.64 + 3.05i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-4.08 - 2.68i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (-5.78 + 1.37i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-6.42 + 8.63i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (5.75 - 9.96i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.19 + 3.97i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-0.105 - 0.245i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (1.71 - 0.860i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (1.17 - 6.65i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 7.81i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (3.60 - 2.37i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (2.12 - 1.39i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.935 - 5.30i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.51 + 6.90i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−9.939535664093958580072418034718, −9.335348447806723506457194085255, −8.730699609532378222340304254402, −8.041155195291113686417627448407, −7.19439385819651277942998553148, −5.89833484241513069411113571878, −5.42961933071146780889815340843, −4.27098171057568408525453090205, −1.95886707126336079649940872581, −1.09290906072308985825210268530,
1.37576718250089518082359742867, 2.30157636298514563204683184911, 3.24163255858891199312223295386, 5.05562500764417990638801265513, 5.99229276299820875142158590323, 7.29153877413190643434102457748, 7.83194711863547546437591432771, 8.947784230182035863768622095545, 9.701008638304053011620881800228, 10.42222833267237874905622711101
