L(s) = 1 | + (0.442 + 0.301i)2-s + (−0.944 + 1.45i)3-s + (−0.625 − 1.59i)4-s + (−0.835 + 3.66i)5-s + (−0.855 + 0.357i)6-s + (1.43 + 2.22i)7-s + (0.442 − 1.93i)8-s + (−1.21 − 2.74i)9-s + (−1.47 + 1.36i)10-s + (−2.52 + 1.21i)11-s + (2.90 + 0.597i)12-s + (−2.75 − 1.87i)13-s + (−0.0356 + 1.41i)14-s + (−4.52 − 4.67i)15-s + (−1.73 + 1.60i)16-s + (−0.492 + 1.25i)17-s + ⋯ |
L(s) = 1 | + (0.312 + 0.213i)2-s + (−0.545 + 0.838i)3-s + (−0.312 − 0.797i)4-s + (−0.373 + 1.63i)5-s + (−0.349 + 0.145i)6-s + (0.542 + 0.840i)7-s + (0.156 − 0.685i)8-s + (−0.405 − 0.914i)9-s + (−0.466 + 0.432i)10-s + (−0.761 + 0.366i)11-s + (0.839 + 0.172i)12-s + (−0.763 − 0.520i)13-s + (−0.00951 + 0.378i)14-s + (−1.16 − 1.20i)15-s + (−0.432 + 0.401i)16-s + (−0.119 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00726203 + 0.688299i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00726203 + 0.688299i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.944 - 1.45i)T \) |
| 7 | \( 1 + (-1.43 - 2.22i)T \) |
good | 2 | \( 1 + (-0.442 - 0.301i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (0.835 - 3.66i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (2.52 - 1.21i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (2.75 + 1.87i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.492 - 1.25i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.71 + 4.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.43 - 5.55i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (4.15 - 0.626i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (3.08 - 5.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.08 + 0.615i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-0.669 + 0.206i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (4.39 + 1.35i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-2.68 - 1.83i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-11.1 - 1.67i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-6.55 - 2.02i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (3.34 - 8.51i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-7.15 + 12.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.427 - 0.536i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.864 - 11.5i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-3.38 - 5.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.6 + 7.93i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (4.99 - 3.40i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (0.364 - 0.630i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.31287853543676489943357446897, −10.66661001860757898597675773124, −9.998348010621010737471026867386, −9.183900339396171279940115899626, −7.68235338128489102220253483139, −6.76094726909356668708939333937, −5.63216096100342577125813296591, −5.12189475296501730965680910793, −3.81046462560657087841928709399, −2.54044094759444409982676990295,
0.40616338950896333918212031643, 2.06345351907615337628387415943, 3.99395141636082155189526258308, 4.79921851019894179861208776932, 5.58791237308968729067346192480, 7.27882693526873556702152929517, 7.982703774247092511804070187828, 8.432700339834087576785658179583, 9.751501573482363269917659841481, 11.09241326103223220326111556761