| L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (1.07 − 0.314i)5-s + (−0.415 − 0.909i)6-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.730 + 0.843i)10-s + (−2.01 − 1.29i)11-s + (0.841 + 0.540i)12-s + (−2.56 + 2.95i)13-s + (0.959 + 0.281i)14-s + (0.158 + 1.10i)15-s + (−0.654 − 0.755i)16-s + (−1.91 − 4.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (0.478 − 0.140i)5-s + (−0.169 − 0.371i)6-s + (−0.247 − 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.230 + 0.266i)10-s + (−0.608 − 0.390i)11-s + (0.242 + 0.156i)12-s + (−0.710 + 0.820i)13-s + (0.256 + 0.0752i)14-s + (0.0409 + 0.285i)15-s + (−0.163 − 0.188i)16-s + (−0.464 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.172509 - 0.250020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.172509 - 0.250020i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.73 - 3.00i)T \) |
| good | 5 | \( 1 + (-1.07 + 0.314i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (2.01 + 1.29i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.56 - 2.95i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.91 + 4.19i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.80 + 6.13i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.74 + 3.81i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.520 - 3.62i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (3.02 + 0.888i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (7.97 - 2.34i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0160 + 0.111i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + (3.20 + 3.70i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-5.92 + 6.83i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.418 - 2.91i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 6.77i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (1.84 - 1.18i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.05 + 2.30i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-9.70 + 11.2i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (8.37 + 2.45i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.04 + 14.2i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (7.68 - 2.25i)T + (81.6 - 52.4i)T^{2} \) |
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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
−9.589137131639416725377931252968, −9.234231126011617526431475196269, −8.153451551717459592737396354938, −7.20636784399532581008216162889, −6.48191905205714661412138523909, −5.31916084745893749229456566881, −4.76516430704650679283052194479, −3.29799456704866539473843544719, −2.05806981585844232991910906710, −0.15973880528540755876048745978,
1.70810108687034217222820124547, 2.55146850707979701904757671855, 3.75131730028310249314028076415, 5.26903345174794925571936668652, 6.07283964428917217125069426783, 6.99722163168461232717493950613, 8.012275890892450051100529481062, 8.383539666493211534773091564921, 9.799476475619426339377663887069, 10.04792516113019134066370532176
