| L(s) = 1 | + (−0.995 + 0.0950i)2-s + (−0.690 − 0.723i)3-s + (0.981 − 0.189i)4-s + (2.27 + 1.17i)5-s + (0.755 + 0.654i)6-s + (2.41 − 1.08i)7-s + (−0.959 + 0.281i)8-s + (−0.0475 + 0.998i)9-s + (−2.37 − 0.950i)10-s + (0.506 − 5.30i)11-s + (−0.814 − 0.580i)12-s + (−6.62 + 0.952i)13-s + (−2.30 + 1.30i)14-s + (−0.720 − 2.45i)15-s + (0.928 − 0.371i)16-s + (−2.05 − 5.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.703 + 0.0672i)2-s + (−0.398 − 0.417i)3-s + (0.490 − 0.0946i)4-s + (1.01 + 0.524i)5-s + (0.308 + 0.267i)6-s + (0.912 − 0.408i)7-s + (−0.339 + 0.0996i)8-s + (−0.0158 + 0.332i)9-s + (−0.750 − 0.300i)10-s + (0.152 − 1.59i)11-s + (−0.235 − 0.167i)12-s + (−1.83 + 0.264i)13-s + (−0.614 + 0.349i)14-s + (−0.186 − 0.633i)15-s + (0.232 − 0.0929i)16-s + (−0.497 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0178 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0178 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.737446 - 0.750754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.737446 - 0.750754i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.995 - 0.0950i)T \) |
| 3 | \( 1 + (0.690 + 0.723i)T \) |
| 7 | \( 1 + (-2.41 + 1.08i)T \) |
| 23 | \( 1 + (-4.39 - 1.91i)T \) |
| good | 5 | \( 1 + (-2.27 - 1.17i)T + (2.90 + 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.506 + 5.30i)T + (-10.8 - 2.08i)T^{2} \) |
| 13 | \( 1 + (6.62 - 0.952i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.05 + 5.92i)T + (-13.3 + 10.5i)T^{2} \) |
| 19 | \( 1 + (-1.34 + 3.87i)T + (-14.9 - 11.7i)T^{2} \) |
| 29 | \( 1 + (0.547 - 0.631i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.89 + 1.67i)T + (27.5 + 14.2i)T^{2} \) |
| 37 | \( 1 + (2.20 + 0.104i)T + (36.8 + 3.51i)T^{2} \) |
| 41 | \( 1 + (-4.50 - 7.01i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (1.39 - 4.76i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (1.06 - 0.614i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.26 + 9.23i)T + (-12.4 - 51.5i)T^{2} \) |
| 59 | \( 1 + (-4.38 + 10.9i)T + (-42.7 - 40.7i)T^{2} \) |
| 61 | \( 1 + (2.51 + 2.39i)T + (2.90 + 60.9i)T^{2} \) |
| 67 | \( 1 + (9.82 - 6.99i)T + (21.9 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-4.65 + 10.1i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (0.394 + 2.04i)T + (-67.7 + 27.1i)T^{2} \) |
| 79 | \( 1 + (-8.77 - 11.1i)T + (-18.6 + 76.7i)T^{2} \) |
| 83 | \( 1 + (-4.42 - 2.84i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.11 - 4.61i)T + (-79.1 + 40.7i)T^{2} \) |
| 97 | \( 1 + (-3.10 + 1.99i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.644730343058772159074652400054, −9.197545772325085128371150886163, −8.036164538856527994430287859740, −7.17500668087579799801768097475, −6.69275800860820677184544729964, −5.50289277900646176330771344253, −4.89684954528236559000903104018, −2.95623860395552280810899447297, −2.06667071841869827026299353668, −0.63012887401514855777011202655,
1.64351588239836115916367831648, 2.29814026080433782563534986996, 4.22892151258000013463394394805, 5.14316237152824698215789071610, 5.73544662202996240451576604052, 7.03441156966908091686360723445, 7.70393690022911898714843244340, 8.945926051284108566690867070333, 9.320667771305029716317208973645, 10.35401850698587667229845062739
