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
−10.05912489920228652889420686160, −9.141714469909070829297239934994, −8.525021085137433672035287909844, −7.59713531202337077748845821664, −7.01191970389120411343838447175, −6.08060121019537470890473339516, −5.13540640090304812414453486834, −4.29234743669144875764850857077, −3.34856335420024081147512409483, −0.47832071229389434722840464672,
1.43767813133883439299441713143, 2.91097444701345467779875168704, 3.52539050461367269789437705557, 4.35379111687008876367745208008, 5.67306342103052140333734160983, 6.85013313110033886458661382427, 7.914223547948687635851857218084, 9.079350682369943597887101130641, 9.768561910530163858313969292452, 10.63632393437820147438600024697