L(s) = 1 | + (2.25 − 0.339i)2-s + (3.05 − 0.943i)4-s + (0.260 − 3.47i)5-s + (1.30 + 2.30i)7-s + (2.46 − 1.18i)8-s + (−0.594 − 7.92i)10-s + (−0.728 + 1.85i)11-s + (−0.725 − 0.909i)13-s + (3.71 + 4.75i)14-s + (−0.128 + 0.0873i)16-s + (−0.130 + 0.120i)17-s + (2.36 − 4.09i)19-s + (−2.48 − 10.8i)20-s + (−1.01 + 4.43i)22-s + (−2.75 − 2.55i)23-s + ⋯ |
L(s) = 1 | + (1.59 − 0.240i)2-s + (1.52 − 0.471i)4-s + (0.116 − 1.55i)5-s + (0.491 + 0.870i)7-s + (0.872 − 0.420i)8-s + (−0.187 − 2.50i)10-s + (−0.219 + 0.559i)11-s + (−0.201 − 0.252i)13-s + (0.993 + 1.27i)14-s + (−0.0320 + 0.0218i)16-s + (−0.0315 + 0.0293i)17-s + (0.542 − 0.940i)19-s + (−0.555 − 2.43i)20-s + (−0.215 + 0.945i)22-s + (−0.574 − 0.533i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.04679 - 1.26850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.04679 - 1.26850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.30 - 2.30i)T \) |
good | 2 | \( 1 + (-2.25 + 0.339i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (-0.260 + 3.47i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (0.728 - 1.85i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (0.725 + 0.909i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.130 - 0.120i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.75 + 2.55i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.03 - 8.90i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.06 - 7.04i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.87 + 1.50i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (4.57 - 2.20i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.63 - 0.787i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.31 + 0.198i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-7.14 + 2.20i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.0395 + 0.528i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (3.38 + 1.04i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.223 - 0.387i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.323 + 1.41i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (13.2 + 1.99i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.42 + 2.47i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.35 + 4.21i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (2.62 + 6.67i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.52338744319048529460749733062, −10.33790623994974556941980585407, −9.022084761370605200272598772859, −8.456926355581134592361090014084, −7.00853469652196018289901676377, −5.71666227034278127550017022827, −4.97571404953387541410120467996, −4.59484080199832722731252714088, −3.01029843594140223642648314854, −1.70771508725620660031871561533,
2.37237856431447303346685683614, 3.48677814599460388787339307153, 4.23690651641089626849899433316, 5.60406753846529712260375950121, 6.35497363561317083810190638654, 7.23852110695276773325721404981, 7.962188612651495326493620754302, 9.854525903554481710593410952384, 10.58083207548779520709854016129, 11.52857851023119673642490284930