L(s) = 1 | + (−1.54 + 0.893i)3-s + (−0.581 + 1.00i)5-s + (2.23 − 1.41i)7-s + (0.0963 − 0.166i)9-s + (−1.13 − 1.96i)11-s + 1.57·13-s − 2.07i·15-s + (5.96 − 3.44i)17-s + (−0.671 − 0.387i)19-s + (−2.20 + 4.18i)21-s + (−1.11 − 0.643i)23-s + (1.82 + 3.15i)25-s − 5.01i·27-s + 8.38i·29-s + (−0.554 − 0.960i)31-s + ⋯ |
L(s) = 1 | + (−0.893 + 0.515i)3-s + (−0.260 + 0.450i)5-s + (0.845 − 0.533i)7-s + (0.0321 − 0.0556i)9-s + (−0.342 − 0.593i)11-s + 0.438·13-s − 0.536i·15-s + (1.44 − 0.834i)17-s + (−0.154 − 0.0889i)19-s + (−0.480 + 0.912i)21-s + (−0.232 − 0.134i)23-s + (0.364 + 0.631i)25-s − 0.965i·27-s + 1.55i·29-s + (−0.0995 − 0.172i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11595 + 0.405940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11595 + 0.405940i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
good | 3 | \( 1 + (1.54 - 0.893i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.581 - 1.00i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.13 + 1.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 + (-5.96 + 3.44i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.671 + 0.387i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.11 + 0.643i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.38iT - 29T^{2} \) |
| 31 | \( 1 + (0.554 + 0.960i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.84 - 3.37i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.99iT - 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 + (2.90 - 5.03i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.317 + 0.183i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.6 + 6.14i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.93 - 6.81i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.01 - 6.94i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (-9.70 + 5.60i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.6 - 6.72i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.15iT - 83T^{2} \) |
| 89 | \( 1 + (2.53 + 1.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.333iT - 97T^{2} \) |
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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
−10.45876965685807276666036026156, −9.614804969281063271088795625214, −8.373579613549466505761893564838, −7.69594548217332218134362273084, −6.77168728235323142151888268066, −5.59335439630373618421223937347, −5.08968200658683184424378144445, −4.00390850051316100094885739654, −2.90068899846103309749193665834, −1.02590356353491396486376734936,
0.878986143204988167137844570532, 2.14159457348248474000485204316, 3.80589657510387802679936575360, 4.90926409279679202015422033974, 5.70948685873341079716378405757, 6.36117201703769287986147258009, 7.66185843386657338385019718168, 8.117214851135408941530844751188, 9.122994873167137171584615182478, 10.15947792729311427203443974359