| L(s) = 1 | + (−0.733 + 1.20i)2-s + (−2.46 + 1.72i)3-s + (−0.923 − 1.77i)4-s + (3.64 − 0.318i)5-s + (−0.279 − 4.25i)6-s + (1.89 + 3.27i)7-s + (2.82 + 0.183i)8-s + (2.07 − 5.70i)9-s + (−2.28 + 4.63i)10-s + (3.77 − 1.01i)11-s + (5.34 + 2.78i)12-s + (2.26 + 1.58i)13-s + (−5.34 − 0.115i)14-s + (−8.44 + 7.08i)15-s + (−2.29 + 3.27i)16-s + (−1.60 + 0.583i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.854i)2-s + (−1.42 + 0.997i)3-s + (−0.461 − 0.886i)4-s + (1.62 − 0.142i)5-s + (−0.114 − 1.73i)6-s + (0.714 + 1.23i)7-s + (0.997 + 0.0649i)8-s + (0.692 − 1.90i)9-s + (−0.723 + 1.46i)10-s + (1.13 − 0.304i)11-s + (1.54 + 0.802i)12-s + (0.628 + 0.440i)13-s + (−1.42 − 0.0309i)14-s + (−2.17 + 1.82i)15-s + (−0.573 + 0.819i)16-s + (−0.388 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.442619 + 0.795349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442619 + 0.795349i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.733 - 1.20i)T \) |
| 19 | \( 1 + (2.00 + 3.87i)T \) |
| good | 3 | \( 1 + (2.46 - 1.72i)T + (1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (-3.64 + 0.318i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.89 - 3.27i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.77 + 1.01i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.26 - 1.58i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.60 - 0.583i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.956 - 0.802i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.49 - 0.697i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-0.632 - 1.09i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.77 + 1.77i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.13 - 6.41i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.245 + 2.80i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (3.53 - 9.71i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.28 - 0.199i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-6.09 + 13.0i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-0.600 - 0.0525i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (4.31 + 9.24i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (8.81 - 10.5i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (6.36 + 1.12i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.26 + 12.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (10.0 + 2.68i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.976 - 5.53i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.06 + 5.66i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−11.60532816366927068403961718748, −11.00826250117339278159021959226, −9.990647234445065418955422445127, −9.173283624307612466619903374138, −8.764113641632910333409173623199, −6.54456987566123631616112718628, −6.09670851332174668001649490572, −5.34600722516078154133267082053, −4.51863907264859004206660744151, −1.61871319655765747851166137562,
1.15232706452590110754096685128, 1.88535548563464234063934940303, 4.20468626837606706540826669919, 5.53074486320293988325044938669, 6.56313385737341232254813360066, 7.34732964198894632690782830672, 8.654021985917860064505074482601, 10.06817563304717690497629873244, 10.49312950853492034427212938446, 11.33889936889663773347796080762
