L(s) = 1 | + (0.5 − 0.866i)2-s + 2.08·3-s + (−0.499 − 0.866i)4-s + 5-s + (1.04 − 1.80i)6-s + (1.21 + 2.11i)7-s − 0.999·8-s + 1.35·9-s + (0.5 − 0.866i)10-s + (1.67 + 2.90i)11-s + (−1.04 − 1.80i)12-s + (1.21 − 2.11i)13-s + 2.43·14-s + 2.08·15-s + (−0.5 + 0.866i)16-s + (−0.367 + 0.635i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + 1.20·3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.425 − 0.737i)6-s + (0.460 + 0.798i)7-s − 0.353·8-s + 0.450·9-s + (0.158 − 0.273i)10-s + (0.505 + 0.875i)11-s + (−0.301 − 0.521i)12-s + (0.338 − 0.585i)13-s + 0.651·14-s + 0.538·15-s + (−0.125 + 0.216i)16-s + (−0.0890 + 0.154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.72159 - 0.868722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.72159 - 0.868722i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (4.02 - 7.12i)T \) |
good | 3 | \( 1 - 2.08T + 3T^{2} \) |
| 7 | \( 1 + (-1.21 - 2.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.67 - 2.90i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 2.11i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.367 - 0.635i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.895 - 1.55i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.08 + 8.80i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.82 - 10.0i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.32 + 2.29i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + (-1.64 + 2.85i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-2.45 - 4.25i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.36 - 2.36i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.52 - 13.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.84 + 11.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.46T + 89T^{2} \) |
| 97 | \( 1 + (-0.453 + 0.785i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.29738865637203166311409000424, −9.504059469615294242121989836594, −8.795925077056036450036586350811, −8.155780650493176364642335100581, −6.91335280179233561571013054284, −5.72762399720329472548134208312, −4.75828445548160373779705997640, −3.57488234180378721836734137404, −2.55813269034314397415309353503, −1.74105711486051530390101199662,
1.62714314150999692156941851691, 3.19126471747173875784542296364, 3.89151888991004303710072380311, 5.11154929521670251157667995002, 6.21421951841233261037573856398, 7.20974874404960822859342272121, 7.928938047395548900657465461466, 8.945396132252271892768526881727, 9.213730616090377159860394683178, 10.57530611764563469796914205761