| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (−0.296 + 0.342i)5-s + (−0.281 + 0.959i)6-s + (2.35 − 1.19i)7-s + (0.415 + 0.909i)8-s + (0.654 + 0.755i)9-s + (0.381 + 0.244i)10-s + (1.25 + 0.180i)11-s + (0.989 + 0.142i)12-s + (−2.29 + 3.57i)13-s + (−1.52 − 2.16i)14-s + (0.412 − 0.188i)15-s + (0.841 − 0.540i)16-s + (0.531 + 0.156i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (−0.132 + 0.153i)5-s + (−0.115 + 0.391i)6-s + (0.891 − 0.452i)7-s + (0.146 + 0.321i)8-s + (0.218 + 0.251i)9-s + (0.120 + 0.0774i)10-s + (0.378 + 0.0544i)11-s + (0.285 + 0.0410i)12-s + (−0.636 + 0.990i)13-s + (−0.406 − 0.578i)14-s + (0.106 − 0.0485i)15-s + (0.210 − 0.135i)16-s + (0.128 + 0.0378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21761 - 0.379827i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21761 - 0.379827i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (-2.35 + 1.19i)T \) |
| 23 | \( 1 + (-4.66 - 1.09i)T \) |
| good | 5 | \( 1 + (0.296 - 0.342i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.25 - 0.180i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.29 - 3.57i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.531 - 0.156i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.02 - 0.299i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.48 - 2.49i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (4.32 - 1.97i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.91 + 2.52i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.666 + 0.577i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.19 - 2.37i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 8.72iT - 47T^{2} \) |
| 53 | \( 1 + (2.10 + 3.27i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-7.59 + 11.8i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.89 - 12.9i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-8.09 + 1.16i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.744 + 5.17i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.51 + 8.57i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (5.23 - 8.13i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-9.54 - 11.0i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.67 + 5.85i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (11.2 - 12.9i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.11102913150145021576989071431, −9.207163022573925541315345958389, −8.372945299902632785755784888264, −7.31091734023644652613519023481, −6.75596808086501198994791049369, −5.31949642239848147153512129975, −4.63187316782657076417977831111, −3.64135246944422391659008864489, −2.18049919042810482491856037507, −1.08949845088257421589445477944,
0.870501186574583933227667481005, 2.68087136423961547557917300385, 4.26519836747058838856668463028, 4.95690110722990572213900265386, 5.73883063807624003653430716477, 6.63373590587285940387792533732, 7.65046541365298511519650779170, 8.337321688269606871327996074369, 9.131682747301914073004691554739, 10.08891875706808010364276542024
