L(s) = 1 | + (−0.890 + 0.585i)2-s + (−0.341 + 0.792i)4-s + (−0.585 + 0.620i)5-s + (0.284 + 0.0332i)7-s + (−0.530 − 3.00i)8-s + (0.158 − 0.896i)10-s + (0.519 − 1.73i)11-s + (−0.325 + 5.58i)13-s + (−0.272 + 0.137i)14-s + (1.04 + 1.11i)16-s + (−4.42 + 1.61i)17-s + (−1.75 − 0.638i)19-s + (−0.291 − 0.676i)20-s + (0.553 + 1.84i)22-s + (−1.34 + 0.157i)23-s + ⋯ |
L(s) = 1 | + (−0.629 + 0.414i)2-s + (−0.170 + 0.396i)4-s + (−0.262 + 0.277i)5-s + (0.107 + 0.0125i)7-s + (−0.187 − 1.06i)8-s + (0.0499 − 0.283i)10-s + (0.156 − 0.522i)11-s + (−0.0901 + 1.54i)13-s + (−0.0729 + 0.0366i)14-s + (0.262 + 0.278i)16-s + (−1.07 + 0.390i)17-s + (−0.402 − 0.146i)19-s + (−0.0652 − 0.151i)20-s + (0.117 + 0.394i)22-s + (−0.280 + 0.0328i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0584509 - 0.167652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0584509 - 0.167652i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.890 - 0.585i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (0.585 - 0.620i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.284 - 0.0332i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.519 + 1.73i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (0.325 - 5.58i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (4.42 - 1.61i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (1.75 + 0.638i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.34 - 0.157i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (5.35 + 2.68i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (2.54 + 3.41i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (5.46 + 4.58i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (9.58 + 6.30i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (-12.3 + 2.93i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 1.64i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (-1.54 + 2.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.361 + 1.20i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-1.09 - 2.54i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (5.61 - 2.81i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (2.24 - 12.7i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.11 + 6.32i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (5.95 - 3.91i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (11.3 - 7.44i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.0750 - 0.425i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.92 + 7.34i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.98551399020905989926445802321, −9.792184809970784250807028081558, −8.940388379812004766504233123760, −8.561800240920141545714244139939, −7.31536113277446880443664201017, −6.91073551933895164965318490413, −5.81826557562837480331738778854, −4.28860755785417684556505391696, −3.66831886749124636691597481456, −2.03341952648700399818625471553,
0.11105255139586935055389527065, 1.63419162339504205580274680054, 2.92506186663719302374755592112, 4.45230378207027191563118229064, 5.25217379661845067988525872469, 6.29050590573060072897389095048, 7.51338907253072408396432414462, 8.375691357196147276515767877713, 9.033707847539300364898026316547, 9.971518195178423997421572330685