L(s) = 1 | + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−2.25 + 1.44i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.381 + 2.65i)10-s + (−0.157 − 0.345i)11-s + (−0.415 − 0.909i)12-s + (0.794 + 5.52i)13-s + (−0.841 − 0.540i)14-s + (−2.57 + 0.754i)15-s + (−0.142 + 0.989i)16-s + (−1.77 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (0.293 − 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−1.00 + 0.647i)5-s + (0.267 − 0.308i)6-s + (0.0537 − 0.374i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.120 + 0.838i)10-s + (−0.0476 − 0.104i)11-s + (−0.119 − 0.262i)12-s + (0.220 + 1.53i)13-s + (−0.224 − 0.144i)14-s + (−0.663 + 0.194i)15-s + (−0.0355 + 0.247i)16-s + (−0.429 + 0.495i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40828 + 0.592736i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40828 + 0.592736i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.24 - 2.24i)T \) |
good | 5 | \( 1 + (2.25 - 1.44i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (0.157 + 0.345i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.794 - 5.52i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.77 - 2.04i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-5.17 - 5.97i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-6.26 + 7.23i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (3.25 - 0.954i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-2.19 - 1.41i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.54 - 0.991i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.33 - 0.391i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + (0.481 - 3.35i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.98 - 13.7i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.24 + 1.83i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (6.56 - 14.3i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-6.21 + 13.6i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (1.72 + 1.99i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.09 - 14.5i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (7.87 + 5.06i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.06 - 0.311i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (0.135 - 0.0867i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.15719722709647879646686332532, −9.493884185837053178710333978919, −8.429281333552707068854588596050, −7.71362664268748835512971983104, −6.84350677607047187490039787488, −5.76399787391133249037571137988, −4.13854213726583374988635723095, −4.03126040080572117166639461450, −2.88685854561332784590419938888, −1.58976400766564066872190649378,
0.63163986083710815921538235574, 2.71538671115392492861147877034, 3.62730772007584593200739432032, 4.75666387114890350310223775738, 5.38386303788879891745075388256, 6.67857051253201278983124342548, 7.53751767907574588675287004036, 8.178068961773188692848488900227, 8.769990604872796592700610679675, 9.611169730016315592103915395494