L(s) = 1 | + (1.65 + 0.517i)3-s + (−2.52 + 0.445i)5-s + (1.40 − 1.67i)7-s + (2.46 + 1.71i)9-s + (−0.751 + 4.26i)11-s + (5.33 + 1.94i)13-s + (−4.41 − 0.572i)15-s + (5.53 − 3.19i)17-s + (2.85 + 1.64i)19-s + (3.19 − 2.04i)21-s + (−3.31 + 2.78i)23-s + (1.49 − 0.545i)25-s + (3.18 + 4.10i)27-s + (0.131 + 0.362i)29-s + (−4.37 − 5.21i)31-s + ⋯ |
L(s) = 1 | + (0.954 + 0.298i)3-s + (−1.13 + 0.199i)5-s + (0.532 − 0.634i)7-s + (0.821 + 0.570i)9-s + (−0.226 + 1.28i)11-s + (1.47 + 0.538i)13-s + (−1.13 − 0.147i)15-s + (1.34 − 0.775i)17-s + (0.654 + 0.378i)19-s + (0.697 − 0.446i)21-s + (−0.691 + 0.580i)23-s + (0.299 − 0.109i)25-s + (0.613 + 0.789i)27-s + (0.0245 + 0.0673i)29-s + (−0.785 − 0.935i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68705 + 0.523542i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68705 + 0.523542i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.65 - 0.517i)T \) |
good | 5 | \( 1 + (2.52 - 0.445i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.40 + 1.67i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.751 - 4.26i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-5.33 - 1.94i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.53 + 3.19i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.85 - 1.64i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.31 - 2.78i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.131 - 0.362i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.37 + 5.21i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.81 + 6.61i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.138 - 0.380i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (9.35 + 1.64i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.74 + 5.65i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 7.00iT - 53T^{2} \) |
| 59 | \( 1 + (0.296 + 1.68i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.66 - 7.26i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.683 + 1.87i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.85 - 3.21i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.37 + 9.31i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.34 - 6.42i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.31 - 0.479i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (6.85 + 3.95i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.09 + 11.8i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.28443712701730952239845053588, −10.21298371369811861012846019504, −9.500862581595203930738582632075, −8.280578968983002741727012761219, −7.65870238856113995156205202916, −7.06770374006734391010573221263, −5.22213816316608004371611425851, −3.99709587458684888691872395251, −3.53170078974778147466644932635, −1.70201540289951393973345379569,
1.26817040130755446162177919607, 3.18267482322682430884824684866, 3.73400628478469041279065715083, 5.30039477768763743443051896345, 6.43264365588052050009757919718, 7.918972840623806277253021711392, 8.220716732810443187392806272746, 8.797659298960346412201132827549, 10.18330553858597210280653740321, 11.20453820749951211318885122074