| L(s) = 1 | + (−0.973 + 0.230i)2-s + (−0.598 − 1.62i)3-s + (0.893 − 0.448i)4-s + (−1.25 − 1.67i)5-s + (0.956 + 1.44i)6-s + (−0.725 + 0.477i)7-s + (−0.766 + 0.642i)8-s + (−2.28 + 1.94i)9-s + (1.60 + 1.34i)10-s + (−4.15 − 0.486i)11-s + (−1.26 − 1.18i)12-s + (−0.644 − 2.15i)13-s + (0.596 − 0.631i)14-s + (−1.98 + 3.03i)15-s + (0.597 − 0.802i)16-s + (0.766 − 4.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 + 0.163i)2-s + (−0.345 − 0.938i)3-s + (0.446 − 0.224i)4-s + (−0.559 − 0.750i)5-s + (0.390 + 0.589i)6-s + (−0.274 + 0.180i)7-s + (−0.270 + 0.227i)8-s + (−0.761 + 0.648i)9-s + (0.507 + 0.425i)10-s + (−1.25 − 0.146i)11-s + (−0.364 − 0.341i)12-s + (−0.178 − 0.597i)13-s + (0.159 − 0.168i)14-s + (−0.511 + 0.784i)15-s + (0.149 − 0.200i)16-s + (0.185 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.115200 - 0.402621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.115200 - 0.402621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.973 - 0.230i)T \) |
| 3 | \( 1 + (0.598 + 1.62i)T \) |
| good | 5 | \( 1 + (1.25 + 1.67i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (0.725 - 0.477i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (4.15 + 0.486i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (0.644 + 2.15i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 4.34i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.162 - 0.921i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (1.33 + 0.876i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (1.00 + 1.06i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.0259 + 0.445i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-6.43 + 2.34i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-12.1 - 2.88i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (4.10 + 9.51i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.603 - 10.3i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (-6.72 + 11.6i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.48 - 0.406i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (9.52 + 4.78i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (-8.97 + 9.50i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (5.72 + 4.80i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.298 + 0.250i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (12.5 - 2.98i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (-1.56 + 0.370i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (-8.48 + 7.12i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.07 + 2.78i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−12.46973587890951066902740047502, −11.52859719393874846669790932308, −10.50705639627610572507250240076, −9.211813005409434812170176325509, −7.991547074943969811513131990194, −7.59664631697393282061145130255, −6.10084657460340959263278417769, −5.02654544100634515596151922130, −2.63752577919122728863237155190, −0.49480758355865515257774302174,
2.89149576318668478118595334983, 4.19107606501831344389353880078, 5.81328812919611735813494082591, 7.12554713709543517743583494987, 8.209044634741029786383589077128, 9.463087988255789681815788952429, 10.38141551581282488342283116618, 10.97069018707506946764561071831, 11.88999690254385834027291881694, 13.10605610190429940022964979306
