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
−13.51584325131699404689004094438, −11.88462706992274407472130483020, −11.04882472685301939819150776636, −9.653526849930033274522022824546, −8.459795628599034679380604719594, −7.47391181943569206089607667172, −7.13031569216565990697910787485, −5.68264100702634578397184381354, −4.71182515970842862430012690508, −2.17405457712729405352603759032,
1.56231304305787179874822128324, 2.92693858040266984397980484194, 4.65201673745463351170716743084, 5.35849353619970803892147036714, 7.72482674598771653726715764717, 8.824359598088697055466236335193, 9.490042332092662784985019204444, 10.77403721238174381707002687460, 11.13888638327683751296235005496, 12.27605691810971723556553276099