| L(s) = 1 | + (1.38 + 0.277i)2-s + (−0.819 + 0.573i)3-s + (1.84 + 0.769i)4-s + (−2.81 + 0.246i)5-s + (−1.29 + 0.568i)6-s + (−2.32 − 4.02i)7-s + (2.34 + 1.57i)8-s + (0.342 − 0.939i)9-s + (−3.97 − 0.438i)10-s + (2.94 − 0.789i)11-s + (−1.95 + 0.428i)12-s + (3.69 + 2.58i)13-s + (−2.10 − 6.22i)14-s + (2.16 − 1.81i)15-s + (2.81 + 2.84i)16-s + (4.88 − 1.77i)17-s + ⋯ |
| L(s) = 1 | + (0.980 + 0.196i)2-s + (−0.472 + 0.331i)3-s + (0.923 + 0.384i)4-s + (−1.25 + 0.110i)5-s + (−0.528 + 0.231i)6-s + (−0.878 − 1.52i)7-s + (0.829 + 0.558i)8-s + (0.114 − 0.313i)9-s + (−1.25 − 0.138i)10-s + (0.888 − 0.238i)11-s + (−0.563 + 0.123i)12-s + (1.02 + 0.718i)13-s + (−0.563 − 1.66i)14-s + (0.558 − 0.468i)15-s + (0.704 + 0.710i)16-s + (1.18 − 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09934 - 0.00268980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09934 - 0.00268980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.277i)T \) |
| 3 | \( 1 + (0.819 - 0.573i)T \) |
| 19 | \( 1 + (-2.84 - 3.30i)T \) |
| good | 5 | \( 1 + (2.81 - 0.246i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (2.32 + 4.02i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.94 + 0.789i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.69 - 2.58i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-4.88 + 1.77i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.11 + 2.61i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.40 + 1.58i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (2.56 + 4.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.40 + 2.40i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.0704 - 0.399i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.866 + 9.89i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.474 + 1.30i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.45 - 0.390i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (2.70 - 5.80i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (5.97 + 0.522i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (0.936 + 2.00i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-7.35 + 8.76i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (4.34 + 0.765i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.08 - 6.17i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-9.19 - 2.46i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.51 - 8.58i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.79 - 15.9i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.48258950488607004950296944119, −9.376135086733693743677225597957, −8.069093440378233331563466109002, −7.23663043455195597839234334448, −6.70034433668986022479522224183, −5.77844164768200642716805545551, −4.45636354703662990831662538651, −3.69494813805985353870351372870, −3.48851939235276950634872505417, −0.972664547331037183012866877769,
1.25043924695044316966501033140, 3.04592266989306023432844881611, 3.54173046919306536723627814263, 4.88144411372172475735492816939, 5.72419045949887367116195931421, 6.43023499416217858745958978551, 7.32142071216400474068200074666, 8.301351394752327327563833591059, 9.265970720502973188424959487257, 10.34040070145338650087368476344
