| L(s) = 1 | + (0.991 + 1.00i)2-s + (0.670 − 1.59i)3-s + (−0.0342 + 1.99i)4-s + (−0.331 + 0.0193i)5-s + (2.27 − 0.907i)6-s + (1.21 − 5.12i)7-s + (−2.05 + 1.94i)8-s + (−2.10 − 2.14i)9-s + (−0.348 − 0.315i)10-s + (3.72 + 2.45i)11-s + (3.17 + 1.39i)12-s + (4.04 + 0.472i)13-s + (6.37 − 3.85i)14-s + (−0.191 + 0.542i)15-s + (−3.99 − 0.136i)16-s + (1.32 + 1.57i)17-s + ⋯ |
| L(s) = 1 | + (0.701 + 0.713i)2-s + (0.386 − 0.922i)3-s + (−0.0171 + 0.999i)4-s + (−0.148 + 0.00863i)5-s + (0.928 − 0.370i)6-s + (0.458 − 1.93i)7-s + (−0.725 + 0.688i)8-s + (−0.700 − 0.713i)9-s + (−0.110 − 0.0996i)10-s + (1.12 + 0.739i)11-s + (0.915 + 0.402i)12-s + (1.12 + 0.131i)13-s + (1.70 − 1.03i)14-s + (−0.0494 + 0.140i)15-s + (−0.999 − 0.0342i)16-s + (0.320 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12890 - 0.0585854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12890 - 0.0585854i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.991 - 1.00i)T \) |
| 3 | \( 1 + (-0.670 + 1.59i)T \) |
| good | 5 | \( 1 + (0.331 - 0.0193i)T + (4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (-1.21 + 5.12i)T + (-6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-3.72 - 2.45i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-4.04 - 0.472i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (-1.32 - 1.57i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (1.89 - 2.25i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (5.76 - 1.36i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (-1.29 + 0.967i)T + (8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (0.754 + 0.225i)T + (25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.491 - 2.78i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-4.71 + 2.03i)T + (28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (-0.199 - 0.396i)T + (-25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-2.48 - 8.28i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (3.62 - 2.09i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.28 - 2.16i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-7.27 + 7.71i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (5.77 + 4.29i)T + (19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 4.03i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (6.53 - 2.37i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.652 - 0.281i)T + (54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (3.61 - 8.38i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (0.902 + 2.47i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.958 - 16.4i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−11.85485583107405840811183020783, −10.97538949812861256461403247798, −9.574960719269015086671887889142, −8.194290988144808888424554676479, −7.70342598237326954544316709137, −6.78598774739599138260618867297, −6.05837060025902723284191024619, −4.16513414178909015694165740736, −3.75435451903815451790639192667, −1.54661424668841550075351682728,
2.11153030577621377772246252273, 3.30634656992079901820788375342, 4.32850039786020498915118624248, 5.59457917562996083797446494314, 6.11388325549038401098924763339, 8.393922954143287112790571528340, 8.919704283465561336243106582211, 9.750313763033939565003275789568, 11.00618115523200063769883114797, 11.58717860442624195182743045695
