| L(s) = 1 | + (0.244 + 1.62i)2-s + (−1.73 + 0.0837i)3-s + (−0.666 + 0.205i)4-s + (−0.532 − 0.363i)5-s + (−0.559 − 2.78i)6-s + (1.40 − 2.24i)7-s + (0.928 + 1.92i)8-s + (2.98 − 0.289i)9-s + (0.459 − 0.953i)10-s + (0.457 + 3.03i)11-s + (1.13 − 0.411i)12-s + (2.65 + 1.04i)13-s + (3.98 + 1.72i)14-s + (0.951 + 0.583i)15-s + (−4.05 + 2.76i)16-s + (−0.255 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.173 + 1.14i)2-s + (−0.998 + 0.0483i)3-s + (−0.333 + 0.102i)4-s + (−0.238 − 0.162i)5-s + (−0.228 − 1.13i)6-s + (0.529 − 0.848i)7-s + (0.328 + 0.681i)8-s + (0.995 − 0.0966i)9-s + (0.145 − 0.301i)10-s + (0.137 + 0.914i)11-s + (0.327 − 0.118i)12-s + (0.735 + 0.288i)13-s + (1.06 + 0.461i)14-s + (0.245 + 0.150i)15-s + (−1.01 + 0.691i)16-s + (−0.0618 − 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.758345 + 1.00842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.758345 + 1.00842i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 - 0.0837i)T \) |
| 7 | \( 1 + (-1.40 + 2.24i)T \) |
| good | 2 | \( 1 + (-0.244 - 1.62i)T + (-1.91 + 0.589i)T^{2} \) |
| 5 | \( 1 + (0.532 + 0.363i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.457 - 3.03i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-2.65 - 1.04i)T + (9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (0.255 + 1.11i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 2.78iT - 19T^{2} \) |
| 23 | \( 1 + (-2.56 - 8.32i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.370 + 1.19i)T + (-23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (-4.77 + 2.75i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.37 - 10.4i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (3.15 + 2.15i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-1.21 + 0.828i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-4.72 + 0.712i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.35 - 0.308i)T + (47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.720 - 9.61i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.916 + 2.97i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (2.45 + 4.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.56 - 1.49i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.705 + 0.562i)T + (16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + (-6.63 + 11.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.16 + 5.51i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (4.79 - 6.00i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (11.1 + 6.42i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.47952168112322645390579344536, −10.56651019469756420353788806508, −9.676759239410038025586346145527, −8.222292371632758703572008499541, −7.44644675653197961491539361077, −6.74345697483800302909433381355, −5.82783523196820875415010404343, −4.81924447203646434030283476366, −4.09214721058622847791471215972, −1.50445205738099838435560220474,
0.993671105277169647104068881662, 2.50402318138854887626695745872, 3.79633374109014225343733010161, 4.93320764781766458392937419820, 6.01828645576601418482912707477, 6.93790971450767195424182148239, 8.272904387747876162641868880184, 9.269410568715324531492227248082, 10.53820134894339883209446964441, 11.04979553793606190023250940689
