L(s) = 1 | + (1.01 + 1.74i)2-s + (−1.70 + 0.311i)3-s + (−1.04 + 1.80i)4-s − 4.18·5-s + (−2.26 − 2.66i)6-s + (−0.976 + 1.69i)7-s − 0.164·8-s + (2.80 − 1.06i)9-s + (−4.22 − 7.32i)10-s + (−0.669 + 1.15i)11-s + (1.21 − 3.39i)12-s + (−0.975 + 1.68i)13-s − 3.94·14-s + (7.13 − 1.30i)15-s + (1.91 + 3.31i)16-s + (−3.34 + 5.78i)17-s + ⋯ |
L(s) = 1 | + (0.714 + 1.23i)2-s + (−0.983 + 0.179i)3-s + (−0.520 + 0.901i)4-s − 1.87·5-s + (−0.925 − 1.08i)6-s + (−0.368 + 0.639i)7-s − 0.0583·8-s + (0.935 − 0.354i)9-s + (−1.33 − 2.31i)10-s + (−0.201 + 0.349i)11-s + (0.349 − 0.980i)12-s + (−0.270 + 0.468i)13-s − 1.05·14-s + (1.84 − 0.336i)15-s + (0.478 + 0.829i)16-s + (−0.810 + 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0729137 - 0.678901i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0729137 - 0.678901i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.70 - 0.311i)T \) |
| 19 | \( 1 + (-4.11 + 1.44i)T \) |
good | 2 | \( 1 + (-1.01 - 1.74i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 7 | \( 1 + (0.976 - 1.69i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.669 - 1.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.975 - 1.68i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.34 - 5.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.986 + 1.70i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 + (0.385 + 0.668i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.26T + 37T^{2} \) |
| 41 | \( 1 - 7.59T + 41T^{2} \) |
| 43 | \( 1 + (3.97 + 6.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.10T + 47T^{2} \) |
| 53 | \( 1 + (-3.75 - 6.49i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 0.332T + 61T^{2} \) |
| 67 | \( 1 + (6.45 - 11.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.81 + 3.14i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.48 - 4.29i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.58 + 4.48i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.30 + 10.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.569 - 0.985i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.87 - 3.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.10039832777523071385657691750, −12.39668874053622867203056530378, −11.54263755925810065342328245647, −10.62099484359646626586438625170, −8.908533964217024877315214320796, −7.64237201265233014268276999222, −6.96343247048078121931262151175, −5.85452019998197360220284591621, −4.65930075292953633173201034855, −3.88307821445618088766634143341,
0.58386483172438276956120003415, 3.19912908138102271888578294809, 4.22121642600204879555252794815, 5.17298410870564508811532078619, 7.11129455285726244736030915639, 7.74219952109247183554849489239, 9.707295044680612506521147155936, 10.94966183691660231861118909540, 11.32744530001886813099496802387, 12.03653587449847624822748283531