| 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
| \(L(1)\) |
\(\approx\) |
\(1.91898 - 0.669039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91898 - 0.669039i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.40794070908707062931707317081, −9.624431203734322927550131141688, −8.430991934264302064480110113064, −7.64456841630827432058653107065, −6.49160928191851245099631975064, −5.76671670435396989429745252453, −5.10430045976643973560522678865, −3.89188883320480170934071549795, −2.80673593017831027859482305436, −0.935123439228029630477333408252,
1.74870239071587169902833011429, 2.90846908545743896871860569117, 4.12724107639862557495842419437, 4.80820383387181737322268365330, 5.78758546050122537484986384080, 7.07616750439746154603899941066, 7.82493508281840361264954730372, 8.992624392936640220756209297374, 9.448119762401165068669775914739, 10.71611230460291544173712076417
