| L(s) = 1 | + (−1.16 − 0.323i)2-s + (−0.467 − 0.282i)4-s + (1.06 − 0.0411i)5-s + (−5.09 − 0.796i)7-s + (2.10 + 2.23i)8-s + (−1.24 − 0.295i)10-s + (−0.460 − 3.36i)11-s + (1.07 − 1.56i)13-s + (5.65 + 2.57i)14-s + (−1.21 − 2.30i)16-s + (1.45 − 0.730i)17-s + (0.362 + 6.23i)19-s + (−0.507 − 0.279i)20-s + (−0.554 + 4.06i)22-s + (−1.67 + 4.33i)23-s + ⋯ |
| L(s) = 1 | + (−0.821 − 0.228i)2-s + (−0.233 − 0.141i)4-s + (0.474 − 0.0183i)5-s + (−1.92 − 0.301i)7-s + (0.744 + 0.789i)8-s + (−0.393 − 0.0932i)10-s + (−0.138 − 1.01i)11-s + (0.298 − 0.434i)13-s + (1.51 + 0.687i)14-s + (−0.304 − 0.577i)16-s + (0.353 − 0.177i)17-s + (0.0832 + 1.42i)19-s + (−0.113 − 0.0625i)20-s + (−0.118 + 0.865i)22-s + (−0.349 + 0.903i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0820 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.243904 + 0.224652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243904 + 0.224652i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.16 + 0.323i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-1.06 + 0.0411i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (5.09 + 0.796i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.460 + 3.36i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 1.56i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 0.730i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.362 - 6.23i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (1.67 - 4.33i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-5.05 - 3.61i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.36 - 3.85i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (3.97 + 5.33i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (1.40 - 5.44i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-3.72 - 3.38i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-3.71 - 4.25i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.178 + 1.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (7.85 - 3.20i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (4.86 - 2.93i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (2.72 - 1.94i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-4.36 - 14.5i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (-3.76 + 0.892i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (9.19 - 9.01i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-1.47 - 5.72i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-0.365 + 1.22i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (11.0 + 0.427i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.25613417754133180412057178969, −9.811983599810337850689215785022, −9.113345466733858899518256293408, −8.234342361197484694247247797015, −7.25599054507636846448840325335, −5.99865426731045609383599750252, −5.60801381013410491635408600039, −3.85269586268414010317539771041, −2.97545345019189576065131419604, −1.23495226827746904417891576440,
0.24982651994966467906470292472, 2.28477357379281491957153827626, 3.56729433612705074216355868005, 4.65889441775433967583027874118, 6.09631298527584732708165810565, 6.75377252747560566270518308139, 7.55033478674608381916220183741, 8.807711618830692745738167375321, 9.316665869936017382668274338645, 9.964017395766527171891005219170
