| 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 | \( 1 + (0.135 + 0.0867i)T + (40.2 + 88.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.611169730016315592103915395494, −8.769990604872796592700610679675, −8.178068961773188692848488900227, −7.53751767907574588675287004036, −6.67857051253201278983124342548, −5.38386303788879891745075388256, −4.75666387114890350310223775738, −3.62730772007584593200739432032, −2.71538671115392492861147877034, −0.63163986083710815921538235574,
1.58976400766564066872190649378, 2.88685854561332784590419938888, 4.03126040080572117166639461450, 4.13854213726583374988635723095, 5.76399787391133249037571137988, 6.84350677607047187490039787488, 7.71362664268748835512971983104, 8.429281333552707068854588596050, 9.493884185837053178710333978919, 10.15719722709647879646686332532
