| L(s) = 1 | + (0.132 + 0.0519i)2-s + (−1.45 − 1.34i)4-s + (1.27 + 0.870i)5-s + (−1.61 − 2.09i)7-s + (−0.245 − 0.510i)8-s + (0.123 + 0.181i)10-s + (−0.796 − 5.28i)11-s + (−4.42 + 3.52i)13-s + (−0.104 − 0.361i)14-s + (0.289 + 3.86i)16-s + (−2.27 − 0.700i)17-s + (2.04 − 1.18i)19-s + (−0.680 − 2.98i)20-s + (0.169 − 0.741i)22-s + (−1.72 − 5.60i)23-s + ⋯ |
| L(s) = 1 | + (0.0936 + 0.0367i)2-s + (−0.725 − 0.673i)4-s + (0.571 + 0.389i)5-s + (−0.609 − 0.792i)7-s + (−0.0868 − 0.180i)8-s + (0.0391 + 0.0574i)10-s + (−0.240 − 1.59i)11-s + (−1.22 + 0.978i)13-s + (−0.0279 − 0.0966i)14-s + (0.0724 + 0.967i)16-s + (−0.550 − 0.169i)17-s + (0.469 − 0.271i)19-s + (−0.152 − 0.667i)20-s + (0.0360 − 0.158i)22-s + (−0.360 − 1.16i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.383680 - 0.725450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.383680 - 0.725450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.61 + 2.09i)T \) |
| good | 2 | \( 1 + (-0.132 - 0.0519i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.27 - 0.870i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.796 + 5.28i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (4.42 - 3.52i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.27 + 0.700i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.04 + 1.18i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.72 + 5.60i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-9.35 + 2.13i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (6.48 + 3.74i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.55 + 1.44i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (0.180 - 0.0868i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (5.56 + 2.68i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (4.15 - 10.5i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-2.89 + 3.12i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-10.3 + 7.06i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-3.33 - 3.59i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (4.48 - 7.77i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.2 - 2.79i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.02 + 1.97i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-1.88 - 3.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.58 + 7.00i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.95 - 0.294i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 8.13iT - 97T^{2} \) |
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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
−10.58714079173134186844360105880, −9.942524286160579545128919690204, −9.220353470911963915459849851399, −8.195005269619353714886408738994, −6.76371983652752542234943550318, −6.20070108096537361971536649567, −5.01134996030790798107161134539, −3.98023933173403810562806018818, −2.53259406732082335202268991789, −0.48854514826216542025477312556,
2.20003632522828534791497739884, 3.40880336461705090751393640347, 4.92564263084278767621675532278, 5.36268914214984304071647314522, 6.91606441364882605355620795527, 7.81855627431475429431461249867, 8.847575410566279046102759353677, 9.727093509630977227095447130910, 10.05573293415503144953051852041, 11.83448104071224776132450255396
