| L(s) = 1 | + (−0.450 − 0.125i)2-s + (−1.52 − 0.920i)4-s + (1.87 − 0.0727i)5-s + (−1.75 − 0.275i)7-s + (1.21 + 1.28i)8-s + (−0.852 − 0.202i)10-s + (−0.122 − 0.894i)11-s + (1.98 − 2.89i)13-s + (0.757 + 0.344i)14-s + (1.27 + 2.42i)16-s + (−1.35 + 0.678i)17-s + (−0.248 − 4.26i)19-s + (−2.92 − 1.61i)20-s + (−0.0570 + 0.417i)22-s + (−0.323 + 0.836i)23-s + ⋯ |
| L(s) = 1 | + (−0.318 − 0.0885i)2-s + (−0.762 − 0.460i)4-s + (0.837 − 0.0325i)5-s + (−0.665 − 0.104i)7-s + (0.428 + 0.454i)8-s + (−0.269 − 0.0638i)10-s + (−0.0368 − 0.269i)11-s + (0.551 − 0.803i)13-s + (0.202 + 0.0920i)14-s + (0.319 + 0.605i)16-s + (−0.327 + 0.164i)17-s + (−0.0570 − 0.979i)19-s + (−0.654 − 0.360i)20-s + (−0.0121 + 0.0891i)22-s + (−0.0673 + 0.174i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.457816 - 0.725657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.457816 - 0.725657i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.450 + 0.125i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-1.87 + 0.0727i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (1.75 + 0.275i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.122 + 0.894i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-1.98 + 2.89i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (1.35 - 0.678i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.248 + 4.26i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (0.323 - 0.836i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (5.35 + 3.82i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-6.83 + 7.83i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (4.40 + 5.92i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (1.34 - 5.21i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-2.48 - 2.25i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (5.78 + 6.62i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (0.445 - 2.52i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.15 + 2.92i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (4.04 - 2.44i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-4.78 + 3.41i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (4.36 + 14.5i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (4.63 - 1.09i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-5.00 + 4.90i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.78 + 6.91i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (1.01 - 3.39i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (4.63 + 0.179i)T + (96.7 + 7.51i)T^{2} \) |
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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
−9.921706562628135931617694322962, −9.444696411047764457895258948354, −8.609821165136678093699215287945, −7.69612108127159547572872552957, −6.29875880663135150424207117240, −5.78195507095716785284494746646, −4.72598917495373134208309255061, −3.51992837045701036476445584282, −2.08691606947212474387568509216, −0.51013883769395071071854490527,
1.62390830457342815034613998165, 3.17673084279204489929598248573, 4.20151715298541248980610021248, 5.30739414734265753875624108047, 6.36007949133018508736444189744, 7.13688619283513833930512235396, 8.355030345952157493722511716281, 8.958274681327156904561569303220, 9.817567431390145135755999568286, 10.23712833583706387094930274684
