L(s) = 1 | + (0.425 − 2.41i)2-s + (−3.75 − 1.36i)4-s + (−1.87 + 1.57i)5-s + (−1.87 + 0.684i)7-s + (−2.44 + 4.24i)8-s + (3.00 + 5.19i)10-s + (1.87 + 1.57i)11-s + (−0.173 − 0.984i)13-s + (0.850 + 4.82i)14-s + (3.06 + 2.57i)16-s + (3.67 + 6.36i)17-s + (0.5 − 0.866i)19-s + (9.20 − 3.35i)20-s + (4.59 − 3.85i)22-s + (2.30 + 0.837i)23-s + ⋯ |
L(s) = 1 | + (0.300 − 1.70i)2-s + (−1.87 − 0.684i)4-s + (−0.839 + 0.704i)5-s + (−0.710 + 0.258i)7-s + (−0.866 + 1.50i)8-s + (0.948 + 1.64i)10-s + (0.565 + 0.474i)11-s + (−0.0481 − 0.273i)13-s + (0.227 + 1.28i)14-s + (0.766 + 0.642i)16-s + (0.891 + 1.54i)17-s + (0.114 − 0.198i)19-s + (2.05 − 0.749i)20-s + (0.979 − 0.822i)22-s + (0.479 + 0.174i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.926052 - 0.108239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.926052 - 0.108239i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.425 + 2.41i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (1.87 - 1.57i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.87 - 0.684i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.87 - 1.57i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.173 + 0.984i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.67 - 6.36i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.30 - 0.837i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.850 - 4.82i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.939 - 0.342i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.850 - 4.82i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-8.42 - 7.07i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (9.20 - 3.35i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + (1.87 - 1.57i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.69 - 1.71i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.21 + 6.89i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (3.67 + 6.36i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.21 - 6.89i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.12 - 12.0i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.36 + 4.49i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.63632393437820147438600024697, −9.768561910530163858313969292452, −9.079350682369943597887101130641, −7.914223547948687635851857218084, −6.85013313110033886458661382427, −5.67306342103052140333734160983, −4.35379111687008876367745208008, −3.52539050461367269789437705557, −2.91097444701345467779875168704, −1.43767813133883439299441713143,
0.47832071229389434722840464672, 3.34856335420024081147512409483, 4.29234743669144875764850857077, 5.13540640090304812414453486834, 6.08060121019537470890473339516, 7.01191970389120411343838447175, 7.59713531202337077748845821664, 8.525021085137433672035287909844, 9.141714469909070829297239934994, 10.05912489920228652889420686160