| 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
−13.10605610190429940022964979306, −11.88999690254385834027291881694, −10.97069018707506946764561071831, −10.38141551581282488342283116618, −9.463087988255789681815788952429, −8.209044634741029786383589077128, −7.12554713709543517743583494987, −5.81328812919611735813494082591, −4.19107606501831344389353880078, −2.89149576318668478118595334983,
0.49480758355865515257774302174, 2.63752577919122728863237155190, 5.02654544100634515596151922130, 6.10084657460340959263278417769, 7.59664631697393282061145130255, 7.991547074943969811513131990194, 9.211813005409434812170176325509, 10.50705639627610572507250240076, 11.52859719393874846669790932308, 12.46973587890951066902740047502
