L(s) = 1 | + (−0.949 − 1.64i)2-s + (−2.65 + 1.40i)3-s + (0.196 − 0.340i)4-s + (4.51 + 2.14i)5-s + (4.82 + 3.02i)6-s + (4.60 − 5.26i)7-s − 8.34·8-s + (5.04 − 7.45i)9-s + (−0.759 − 9.46i)10-s + (−8.35 − 4.82i)11-s + (−0.0423 + 1.17i)12-s − 16.9i·13-s + (−13.0 − 2.57i)14-s + (−14.9 + 0.660i)15-s + (7.13 + 12.3i)16-s + (12.1 − 21.1i)17-s + ⋯ |
L(s) = 1 | + (−0.474 − 0.822i)2-s + (−0.883 + 0.468i)3-s + (0.0490 − 0.0850i)4-s + (0.903 + 0.429i)5-s + (0.804 + 0.504i)6-s + (0.658 − 0.752i)7-s − 1.04·8-s + (0.560 − 0.827i)9-s + (−0.0759 − 0.946i)10-s + (−0.759 − 0.438i)11-s + (−0.00353 + 0.0981i)12-s − 1.30i·13-s + (−0.931 − 0.184i)14-s + (−0.999 + 0.0440i)15-s + (0.446 + 0.772i)16-s + (0.716 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.581838 - 0.722274i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.581838 - 0.722274i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.65 - 1.40i)T \) |
| 5 | \( 1 + (-4.51 - 2.14i)T \) |
| 7 | \( 1 + (-4.60 + 5.26i)T \) |
good | 2 | \( 1 + (0.949 + 1.64i)T + (-2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (8.35 + 4.82i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 16.9iT - 169T^{2} \) |
| 17 | \( 1 + (-12.1 + 21.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.95 - 12.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.354 - 0.614i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 16.5iT - 841T^{2} \) |
| 31 | \( 1 + (-7.12 + 12.3i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-1.08 + 0.626i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 24.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 57.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-16.0 - 27.8i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (8.67 - 15.0i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (75.8 + 43.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-52.2 - 90.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (86.1 + 49.7i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 50.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-81.5 - 47.0i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (3.71 + 6.43i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 69.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-78.3 + 45.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 90.4iT - 9.40e3T^{2} \) |
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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
−13.03127657293516947431383602056, −11.73777808946155529057554645918, −10.86477885617987862970313168335, −10.24822162539249666561464349281, −9.565133945139789922722913564313, −7.70986115414742405220173196974, −6.07610651827143474096994998158, −5.18382668348681626060761930987, −3.02602863730993853941597758306, −0.948914907403487821616535923039,
1.97728881988840027011974878174, 5.03103233399995265292139218946, 5.96324009444984287587291229749, 7.02983178865043003558138267563, 8.228555575400302036792194771177, 9.253381051632133109302830510428, 10.62819121840921634929061655709, 11.97407342480196223726217203068, 12.54900763045933829940020548976, 13.82118125595232408699785560061