| L(s) = 1 | + (1.27 + 1.88i)2-s + (0.789 + 1.54i)3-s + (−1.16 + 2.93i)4-s + (−0.346 − 0.749i)5-s + (−1.89 + 3.45i)6-s + (0.990 − 3.56i)7-s + (−2.57 + 0.566i)8-s + (−1.75 + 2.43i)9-s + (0.967 − 1.60i)10-s + (−1.13 + 1.07i)11-s + (−5.44 + 0.516i)12-s + (−0.644 − 5.93i)13-s + (7.97 − 2.68i)14-s + (0.881 − 1.12i)15-s + (0.240 + 0.228i)16-s + (−2.93 + 0.815i)17-s + ⋯ |
| L(s) = 1 | + (0.901 + 1.32i)2-s + (0.456 + 0.889i)3-s + (−0.584 + 1.46i)4-s + (−0.155 − 0.335i)5-s + (−0.772 + 1.40i)6-s + (0.374 − 1.34i)7-s + (−0.910 + 0.200i)8-s + (−0.584 + 0.811i)9-s + (0.305 − 0.508i)10-s + (−0.343 + 0.325i)11-s + (−1.57 + 0.148i)12-s + (−0.178 − 1.64i)13-s + (2.13 − 0.717i)14-s + (0.227 − 0.290i)15-s + (0.0602 + 0.0570i)16-s + (−0.712 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.979418 + 1.65097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.979418 + 1.65097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.789 - 1.54i)T \) |
| 59 | \( 1 + (-4.93 + 5.88i)T \) |
| good | 2 | \( 1 + (-1.27 - 1.88i)T + (-0.740 + 1.85i)T^{2} \) |
| 5 | \( 1 + (0.346 + 0.749i)T + (-3.23 + 3.81i)T^{2} \) |
| 7 | \( 1 + (-0.990 + 3.56i)T + (-5.99 - 3.60i)T^{2} \) |
| 11 | \( 1 + (1.13 - 1.07i)T + (0.595 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.644 + 5.93i)T + (-12.6 + 2.79i)T^{2} \) |
| 17 | \( 1 + (2.93 - 0.815i)T + (14.5 - 8.76i)T^{2} \) |
| 19 | \( 1 + (2.13 - 1.62i)T + (5.08 - 18.3i)T^{2} \) |
| 23 | \( 1 + (0.526 - 0.993i)T + (-12.9 - 19.0i)T^{2} \) |
| 29 | \( 1 + (-5.88 - 3.98i)T + (10.7 + 26.9i)T^{2} \) |
| 31 | \( 1 + (2.13 - 2.80i)T + (-8.29 - 29.8i)T^{2} \) |
| 37 | \( 1 + (1.21 - 5.50i)T + (-33.5 - 15.5i)T^{2} \) |
| 41 | \( 1 + (7.37 - 3.90i)T + (23.0 - 33.9i)T^{2} \) |
| 43 | \( 1 + (-2.77 + 2.92i)T + (-2.32 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-4.74 - 2.19i)T + (30.4 + 35.8i)T^{2} \) |
| 53 | \( 1 + (6.69 + 11.1i)T + (-24.8 + 46.8i)T^{2} \) |
| 61 | \( 1 + (-1.44 + 0.982i)T + (22.5 - 56.6i)T^{2} \) |
| 67 | \( 1 + (-1.28 - 5.84i)T + (-60.8 + 28.1i)T^{2} \) |
| 71 | \( 1 + (-5.62 + 12.1i)T + (-45.9 - 54.1i)T^{2} \) |
| 73 | \( 1 + (4.28 + 12.7i)T + (-58.1 + 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.598 - 11.0i)T + (-78.5 + 8.54i)T^{2} \) |
| 83 | \( 1 + (0.0783 + 0.477i)T + (-78.6 + 26.5i)T^{2} \) |
| 89 | \( 1 + (8.74 - 12.8i)T + (-32.9 - 82.6i)T^{2} \) |
| 97 | \( 1 + (-1.72 + 5.11i)T + (-77.2 - 58.7i)T^{2} \) |
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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
−13.37576919213333628660473986061, −12.56141470751740258682291342555, −10.73750784987917003835592009214, −10.18670358749740059823079364232, −8.409887751572525372274307923207, −7.912375801648043593669808848455, −6.73032985882153378049090033325, −5.18900049594553331317486788100, −4.55665785618629289261842401727, −3.43211356688544897576053596498,
2.00733387339007049176287904022, 2.75502354145465511364388627824, 4.32855152113809446749279565494, 5.73353838876179539793183633810, 6.99465284359905783884229982453, 8.552556048470197028503641141134, 9.349755611164685695211435962745, 10.95103959167195291944439334307, 11.68208421437728475667086932324, 12.24744048807680595334721914972
