L(s) = 1 | + (−0.335 − 2.04i)2-s + (0.267 + 0.963i)3-s + (−2.17 + 0.733i)4-s + (0.939 − 1.38i)5-s + (1.88 − 0.870i)6-s + (4.70 + 1.03i)7-s + (0.288 + 0.543i)8-s + (−0.856 + 0.515i)9-s + (−3.15 − 1.45i)10-s + (−1.41 − 1.07i)11-s + (−1.28 − 1.90i)12-s + (−3.26 − 1.96i)13-s + (0.540 − 9.96i)14-s + (1.58 + 0.534i)15-s + (−2.64 + 2.00i)16-s + (−1.03 + 0.227i)17-s + ⋯ |
L(s) = 1 | + (−0.237 − 1.44i)2-s + (0.154 + 0.556i)3-s + (−1.08 + 0.366i)4-s + (0.420 − 0.619i)5-s + (0.768 − 0.355i)6-s + (1.77 + 0.391i)7-s + (0.101 + 0.192i)8-s + (−0.285 + 0.171i)9-s + (−0.996 − 0.460i)10-s + (−0.427 − 0.325i)11-s + (−0.372 − 0.548i)12-s + (−0.904 − 0.544i)13-s + (0.144 − 2.66i)14-s + (0.409 + 0.138i)15-s + (−0.660 + 0.502i)16-s + (−0.250 + 0.0551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.810509 - 0.909473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.810509 - 0.909473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.267 - 0.963i)T \) |
| 59 | \( 1 + (6.99 + 3.16i)T \) |
good | 2 | \( 1 + (0.335 + 2.04i)T + (-1.89 + 0.638i)T^{2} \) |
| 5 | \( 1 + (-0.939 + 1.38i)T + (-1.85 - 4.64i)T^{2} \) |
| 7 | \( 1 + (-4.70 - 1.03i)T + (6.35 + 2.93i)T^{2} \) |
| 11 | \( 1 + (1.41 + 1.07i)T + (2.94 + 10.5i)T^{2} \) |
| 13 | \( 1 + (3.26 + 1.96i)T + (6.08 + 11.4i)T^{2} \) |
| 17 | \( 1 + (1.03 - 0.227i)T + (15.4 - 7.13i)T^{2} \) |
| 19 | \( 1 + (-3.52 - 0.383i)T + (18.5 + 4.08i)T^{2} \) |
| 23 | \( 1 + (-2.75 + 3.24i)T + (-3.72 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.14 - 6.96i)T + (-27.4 - 9.25i)T^{2} \) |
| 31 | \( 1 + (7.28 - 0.792i)T + (30.2 - 6.66i)T^{2} \) |
| 37 | \( 1 + (-1.92 + 3.63i)T + (-20.7 - 30.6i)T^{2} \) |
| 41 | \( 1 + (-4.11 - 4.84i)T + (-6.63 + 40.4i)T^{2} \) |
| 43 | \( 1 + (5.34 - 4.06i)T + (11.5 - 41.4i)T^{2} \) |
| 47 | \( 1 + (-3.53 - 5.22i)T + (-17.3 + 43.6i)T^{2} \) |
| 53 | \( 1 + (7.36 - 3.40i)T + (34.3 - 40.3i)T^{2} \) |
| 61 | \( 1 + (1.03 + 6.29i)T + (-57.8 + 19.4i)T^{2} \) |
| 67 | \( 1 + (-5.07 - 9.56i)T + (-37.5 + 55.4i)T^{2} \) |
| 71 | \( 1 + (7.41 + 10.9i)T + (-26.2 + 65.9i)T^{2} \) |
| 73 | \( 1 + (0.132 - 2.44i)T + (-72.5 - 7.89i)T^{2} \) |
| 79 | \( 1 + (-4.04 + 14.5i)T + (-67.6 - 40.7i)T^{2} \) |
| 83 | \( 1 + (10.8 - 10.3i)T + (4.49 - 82.8i)T^{2} \) |
| 89 | \( 1 + (-0.177 + 1.08i)T + (-84.3 - 28.4i)T^{2} \) |
| 97 | \( 1 + (-0.525 - 9.69i)T + (-96.4 + 10.4i)T^{2} \) |
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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
−12.27605691810971723556553276099, −11.13888638327683751296235005496, −10.77403721238174381707002687460, −9.490042332092662784985019204444, −8.824359598088697055466236335193, −7.72482674598771653726715764717, −5.35849353619970803892147036714, −4.65201673745463351170716743084, −2.92693858040266984397980484194, −1.56231304305787179874822128324,
2.17405457712729405352603759032, 4.71182515970842862430012690508, 5.68264100702634578397184381354, 7.13031569216565990697910787485, 7.47391181943569206089607667172, 8.459795628599034679380604719594, 9.653526849930033274522022824546, 11.04882472685301939819150776636, 11.88462706992274407472130483020, 13.51584325131699404689004094438