L(s) = 1 | + (−0.568 − 0.223i)2-s + (−1.19 − 1.10i)4-s + (2.30 + 1.57i)5-s + (0.0953 + 2.64i)7-s + (0.961 + 1.99i)8-s + (−0.962 − 1.41i)10-s + (−0.355 − 2.35i)11-s + (−2.96 + 2.36i)13-s + (0.536 − 1.52i)14-s + (0.141 + 1.89i)16-s + (2.98 + 0.920i)17-s + (−5.52 + 3.19i)19-s + (−1.01 − 4.43i)20-s + (−0.324 + 1.41i)22-s + (2.41 + 7.82i)23-s + ⋯ |
L(s) = 1 | + (−0.402 − 0.157i)2-s + (−0.596 − 0.553i)4-s + (1.03 + 0.704i)5-s + (0.0360 + 0.999i)7-s + (0.339 + 0.705i)8-s + (−0.304 − 0.446i)10-s + (−0.107 − 0.710i)11-s + (−0.823 + 0.656i)13-s + (0.143 − 0.407i)14-s + (0.0354 + 0.473i)16-s + (0.724 + 0.223i)17-s + (−1.26 + 0.732i)19-s + (−0.226 − 0.991i)20-s + (−0.0690 + 0.302i)22-s + (0.503 + 1.63i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.930047 + 0.487388i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.930047 + 0.487388i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.0953 - 2.64i)T \) |
good | 2 | \( 1 + (0.568 + 0.223i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.30 - 1.57i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.355 + 2.35i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.96 - 2.36i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.98 - 0.920i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (5.52 - 3.19i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 7.82i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-8.18 + 1.86i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.00 - 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.29 + 4.91i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (2.49 - 1.20i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-8.98 - 4.32i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.52 - 3.87i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (0.503 - 0.542i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (2.77 - 1.89i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (8.06 + 8.69i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (1.63 - 2.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.32 + 0.986i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (8.43 - 3.31i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (5.27 + 9.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.22 + 7.81i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-11.8 - 1.78i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 3.74iT - 97T^{2} \) |
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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
−11.01605946197846254908795826778, −10.18730674762456524779961765846, −9.548639866259896983599880230721, −8.811828276245095389642814204538, −7.79402360735817770153631634885, −6.20338225297366484465517807526, −5.82074413399790728608887366223, −4.63805022133255188238432205817, −2.84791913811133379726143194511, −1.71851876568059155488229006914,
0.807241237823564957374814325872, 2.68451857074439758615886549264, 4.43239474317687293807027830062, 4.91544743571407380196460955171, 6.47391306187165990702237569969, 7.41386109962094140753269546259, 8.344536403467065093389640704412, 9.167671790945472340513035922368, 10.10553535956581245688949766412, 10.45485433319408908985302013272