L(s) = 1 | + (1.22 + 1.22i)3-s + (−4.67 − 1.77i)5-s + (0.550 − 0.550i)7-s + 2.99i·9-s − 1.55·11-s + (−9.55 − 9.55i)13-s + (−3.55 − 7.89i)15-s + (11.1 − 11.1i)17-s + 12.6i·19-s + 1.34·21-s + (21.3 + 21.3i)23-s + (18.6 + 16.5i)25-s + (−3.67 + 3.67i)27-s + 44.0i·29-s + 44.4·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.934 − 0.355i)5-s + (0.0786 − 0.0786i)7-s + 0.333i·9-s − 0.140·11-s + (−0.734 − 0.734i)13-s + (−0.236 − 0.526i)15-s + (0.655 − 0.655i)17-s + 0.668i·19-s + 0.0642·21-s + (0.928 + 0.928i)23-s + (0.747 + 0.663i)25-s + (−0.136 + 0.136i)27-s + 1.51i·29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.390062596\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390062596\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (4.67 + 1.77i)T \) |
good | 7 | \( 1 + (-0.550 + 0.550i)T - 49iT^{2} \) |
| 11 | \( 1 + 1.55T + 121T^{2} \) |
| 13 | \( 1 + (9.55 + 9.55i)T + 169iT^{2} \) |
| 17 | \( 1 + (-11.1 + 11.1i)T - 289iT^{2} \) |
| 19 | \( 1 - 12.6iT - 361T^{2} \) |
| 23 | \( 1 + (-21.3 - 21.3i)T + 529iT^{2} \) |
| 29 | \( 1 - 44.0iT - 841T^{2} \) |
| 31 | \( 1 - 44.4T + 961T^{2} \) |
| 37 | \( 1 + (20.6 - 20.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 48.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (36.2 + 36.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (42.5 - 42.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-54.4 - 54.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 47.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-81.2 + 81.2i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 87.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-75.9 - 75.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 97.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-41.0 - 41.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (37 - 37i)T - 9.40e3iT^{2} \) |
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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
−10.03352004163376709716692945979, −9.145870786956793083963778386204, −8.290504833226760549309737390404, −7.66466111985554167253265979292, −6.88465365547743098197119209893, −5.30304295130584260847197891458, −4.85020153068870057867500992671, −3.58609744596121770340230158755, −2.92529594646162256639181546879, −1.14307744541407147017863664111,
0.48068709561520182112917469903, 2.15138117447380221104731746403, 3.16089129563756653645597965011, 4.19918230839065001623890438774, 5.13280184757798981923963724751, 6.64032425895462716265760699729, 6.97693506249017562056894995417, 8.175672932120533677620152261338, 8.432751902102058923544688103634, 9.691741137858033650787315678220