| L(s) = 1 | + (1.60 − 0.771i)2-s + (−0.884 + 1.48i)3-s + (0.722 − 0.905i)4-s + (−2.59 − 2.41i)5-s + (−0.267 + 3.06i)6-s + (−1.80 − 1.93i)7-s + (−0.332 + 1.45i)8-s + (−1.43 − 2.63i)9-s + (−6.02 − 1.85i)10-s + (−1.22 − 0.835i)11-s + (0.710 + 1.87i)12-s + (−1.91 − 1.30i)13-s + (−4.38 − 1.70i)14-s + (5.88 − 1.73i)15-s + (1.10 + 4.85i)16-s + (0.421 − 0.0635i)17-s + ⋯ |
| L(s) = 1 | + (1.13 − 0.545i)2-s + (−0.510 + 0.859i)3-s + (0.361 − 0.452i)4-s + (−1.16 − 1.07i)5-s + (−0.109 + 1.25i)6-s + (−0.681 − 0.731i)7-s + (−0.117 + 0.515i)8-s + (−0.478 − 0.877i)9-s + (−1.90 − 0.587i)10-s + (−0.369 − 0.251i)11-s + (0.204 + 0.541i)12-s + (−0.530 − 0.361i)13-s + (−1.17 − 0.456i)14-s + (1.52 − 0.448i)15-s + (0.276 + 1.21i)16-s + (0.102 − 0.0154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.137361 - 0.653864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137361 - 0.653864i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.884 - 1.48i)T \) |
| 7 | \( 1 + (1.80 + 1.93i)T \) |
| good | 2 | \( 1 + (-1.60 + 0.771i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (2.59 + 2.41i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.835i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (1.91 + 1.30i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.421 + 0.0635i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (2.91 + 5.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.675 - 1.72i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-8.25 + 1.24i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 + (1.11 - 2.84i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.34 + 0.722i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (10.4 + 3.22i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-2.64 + 1.27i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (3.69 + 9.42i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-0.506 - 2.21i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-3.38 - 4.24i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 4.60T + 67T^{2} \) |
| 71 | \( 1 + (-9.09 + 11.4i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-4.11 + 2.80i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + 5.71T + 79T^{2} \) |
| 83 | \( 1 + (7.97 - 5.43i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (0.179 + 2.39i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (2.09 - 3.63i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.03582990240609651879803452784, −10.14125002627304195352881369243, −8.971457190903755268160145605398, −8.131310455094871295413455360146, −6.76040909924570338327390756888, −5.34138313010284220750918667106, −4.71500393592897701123904931979, −3.93891303183734963077662646466, −3.10158559513145568650073735222, −0.29713542949960614761441561504,
2.59685939707286689714296616726, 3.68720477844618290248903423845, 4.93390828124304135995645214176, 6.03105913100966020597773768748, 6.71730148146179908828724728101, 7.36808587163436083656399575315, 8.358339629144591776018421677329, 9.935123515095880385763529096539, 10.90706549100112217409385432585, 11.93413011296112531730763628436
