| L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0807 + 1.07i)5-s + (−2.64 − 0.115i)7-s + (−0.900 − 0.433i)8-s + (0.0807 − 1.07i)10-s + (−0.249 − 0.634i)11-s + (0.00686 − 0.00860i)13-s + (2.59 + 0.507i)14-s + (0.826 + 0.563i)16-s + (−0.289 − 0.268i)17-s + (−3.12 − 5.41i)19-s + (−0.240 + 1.05i)20-s + (0.151 + 0.664i)22-s + (4.89 − 4.54i)23-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (0.477 + 0.147i)4-s + (0.0361 + 0.482i)5-s + (−0.999 − 0.0435i)7-s + (−0.318 − 0.153i)8-s + (0.0255 − 0.340i)10-s + (−0.0751 − 0.191i)11-s + (0.00190 − 0.00238i)13-s + (0.693 + 0.135i)14-s + (0.206 + 0.140i)16-s + (−0.0701 − 0.0650i)17-s + (−0.717 − 1.24i)19-s + (−0.0537 + 0.235i)20-s + (0.0323 + 0.141i)22-s + (1.02 − 0.947i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.625226 - 0.457084i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.625226 - 0.457084i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.115i)T \) |
| good | 5 | \( 1 + (-0.0807 - 1.07i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.249 + 0.634i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.00686 + 0.00860i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.289 + 0.268i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.12 + 5.41i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.89 + 4.54i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.591 - 2.59i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.87 + 4.98i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.53 - 0.472i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (7.12 + 3.42i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.84 + 3.77i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-11.1 - 1.68i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (7.53 + 2.32i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.377 + 5.03i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-5.40 + 1.66i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (3.11 - 5.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.529 - 2.31i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.11 - 0.318i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (5.18 + 8.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.01 + 2.52i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-5.92 + 15.0i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−10.01997046276562479864982528424, −9.039415944382604777728985143890, −8.590211704761564091385270068526, −7.19970606167267508034583406756, −6.80521357303447560017833590420, −5.88060374905878755471262868262, −4.51880146148175730403150090289, −3.19198813206124848555851958566, −2.43586637073774738752453867890, −0.51337866502732418738856108792,
1.24031805630479915590097084242, 2.72139830418638999303329764373, 3.86630336219647591220140073411, 5.18224521666560442548256016639, 6.14193205625649558434011559306, 6.93417689526299808662550272118, 7.85647090555932365763866635430, 8.788324212244692724200010198295, 9.367273716746799826185525622453, 10.20451192143013775198868470678
