| L(s) = 1 | + (0.833 − 2.28i)2-s + (1.56 + 2.56i)3-s + (−1.48 − 1.24i)4-s + (−8.90 − 1.56i)5-s + (7.16 − 1.44i)6-s + (−1.32 + 1.11i)7-s + (4.35 − 2.51i)8-s + (−4.11 + 8.00i)9-s + (−11.0 + 19.0i)10-s + (8.66 − 1.52i)11-s + (0.869 − 5.74i)12-s + (4.31 − 1.57i)13-s + (1.44 + 3.96i)14-s + (−9.89 − 25.2i)15-s + (−3.47 − 19.6i)16-s + (−5.16 − 2.98i)17-s + ⋯ |
| L(s) = 1 | + (0.416 − 1.14i)2-s + (0.520 + 0.853i)3-s + (−0.370 − 0.311i)4-s + (−1.78 − 0.313i)5-s + (1.19 − 0.240i)6-s + (−0.189 + 0.158i)7-s + (0.544 − 0.314i)8-s + (−0.457 + 0.889i)9-s + (−1.10 + 1.90i)10-s + (0.788 − 0.138i)11-s + (0.0724 − 0.478i)12-s + (0.332 − 0.120i)13-s + (0.103 + 0.283i)14-s + (−0.659 − 1.68i)15-s + (−0.217 − 1.23i)16-s + (−0.304 − 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04709 - 0.406979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04709 - 0.406979i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.56 - 2.56i)T \) |
| good | 2 | \( 1 + (-0.833 + 2.28i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (8.90 + 1.56i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (1.32 - 1.11i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-8.66 + 1.52i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-4.31 + 1.57i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (5.16 + 2.98i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (0.342 + 0.593i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.55 - 10.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (5.73 - 15.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (20.0 + 16.8i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-22.1 + 38.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-23.5 - 64.5i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (0.142 + 0.805i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (0.755 + 0.900i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 46.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (48.2 + 8.51i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (46.4 - 38.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-85.1 + 31.0i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-51.0 - 29.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (15.7 + 27.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (99.2 + 36.1i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-50.5 + 138. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-2.88 + 1.66i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-15.0 - 85.5i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−16.50097432902192303972969795957, −15.78835295592417446883308971038, −14.54710227269057610328104774652, −12.93731656331204619882134497181, −11.67499512432465711045740731347, −10.96445072311416428053658560137, −9.234442587492667376533391135283, −7.74861597988520061509586744756, −4.39167647735238716486069855887, −3.41357251968246831989039095861,
3.96846504996240448051519862456, 6.53983233447434020252870815731, 7.47578717836839085902384681120, 8.520630593858072071380667654624, 11.22447499619343015503788936221, 12.44515569715350319633409913093, 13.96838490192263041016698776927, 14.92814435503727244963783834840, 15.74034832935604589883228089998, 16.96377872496331911240563680471
