L(s) = 1 | + (1.67 − 2.90i)2-s + (−3.62 − 6.28i)4-s − 0.888i·5-s + (−3.29 − 6.17i)7-s − 10.9·8-s + (−2.58 − 1.49i)10-s + 5.50·11-s + (−12.3 − 7.11i)13-s + (−23.4 − 0.792i)14-s + (−3.81 + 6.61i)16-s + (11.2 + 6.51i)17-s + (−22.7 + 13.1i)19-s + (−5.58 + 3.22i)20-s + (9.23 − 16.0i)22-s + 36.0·23-s + ⋯ |
L(s) = 1 | + (0.838 − 1.45i)2-s + (−0.907 − 1.57i)4-s − 0.177i·5-s + (−0.470 − 0.882i)7-s − 1.36·8-s + (−0.258 − 0.149i)10-s + 0.500·11-s + (−0.948 − 0.547i)13-s + (−1.67 − 0.0566i)14-s + (−0.238 + 0.413i)16-s + (0.664 + 0.383i)17-s + (−1.19 + 0.690i)19-s + (−0.279 + 0.161i)20-s + (0.419 − 0.727i)22-s + 1.56·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.190584 - 1.99543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.190584 - 1.99543i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (3.29 + 6.17i)T \) |
good | 2 | \( 1 + (-1.67 + 2.90i)T + (-2 - 3.46i)T^{2} \) |
| 5 | \( 1 + 0.888iT - 25T^{2} \) |
| 11 | \( 1 - 5.50T + 121T^{2} \) |
| 13 | \( 1 + (12.3 + 7.11i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-11.2 - 6.51i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (22.7 - 13.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 - 36.0T + 529T^{2} \) |
| 29 | \( 1 + (16.4 + 28.5i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-32.8 + 18.9i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-21.4 - 37.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-51.1 - 29.5i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (14.4 + 25.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-10.8 - 6.27i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-4.88 + 8.45i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (21.3 - 12.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (15.7 + 9.08i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (15.1 + 26.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 43.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-23.5 - 13.6i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-33.7 + 58.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-8.30 + 4.79i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (54.8 - 31.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (76.9 - 44.4i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.03161865877955691048035116287, −10.93694106473160598196140230671, −10.21989650157777028056803947031, −9.411233029523923181580221740184, −7.80404445976022495812949334863, −6.33624960017093720873222503073, −4.88674630799738462985698026045, −3.92813936371698117874396790314, −2.73459806145355769518785896675, −0.972309915953531180431083759645,
2.90967927045836332882834587566, 4.50386922427814362450066469574, 5.45417847087443495379847141966, 6.62335643623426532037365093774, 7.19922661484645652612375716310, 8.632688101197147943206345328140, 9.369100682947073511576202581789, 10.99512311623181533377829965312, 12.38792895505990215791039269192, 12.81250732672575829814188231175