L(s) = 1 | + (0.894 − 1.54i)3-s + (0.5 − 0.866i)5-s + 0.144·7-s + (−0.0989 − 0.171i)9-s + 5.35·11-s + (0.139 + 0.242i)13-s + (−0.894 − 1.54i)15-s + (−1.02 + 1.77i)17-s + (4.30 + 0.682i)19-s + (0.129 − 0.223i)21-s + (−1.98 − 3.44i)23-s + (−0.499 − 0.866i)25-s + 5.01·27-s + (1.69 + 2.94i)29-s + 4.97·31-s + ⋯ |
L(s) = 1 | + (0.516 − 0.894i)3-s + (0.223 − 0.387i)5-s + 0.0545·7-s + (−0.0329 − 0.0571i)9-s + 1.61·11-s + (0.0388 + 0.0672i)13-s + (−0.230 − 0.399i)15-s + (−0.248 + 0.429i)17-s + (0.987 + 0.156i)19-s + (0.0281 − 0.0488i)21-s + (−0.414 − 0.718i)23-s + (−0.0999 − 0.173i)25-s + 0.964·27-s + (0.315 + 0.546i)29-s + 0.894·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.444469530\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.444469530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4.30 - 0.682i)T \) |
good | 3 | \( 1 + (-0.894 + 1.54i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 0.144T + 7T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 + (-0.139 - 0.242i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.02 - 1.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.98 + 3.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.69 - 2.94i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 + 3.72T + 37T^{2} \) |
| 41 | \( 1 + (0.927 - 1.60i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.43 - 2.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.245 - 0.425i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.83 + 4.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.56 + 6.17i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.22 + 5.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.19 - 8.99i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.35 + 4.08i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.17 + 5.49i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.66 + 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + (-2.92 - 5.07i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.74 - 11.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.236335572994222876089897099417, −8.414339027907630574558356923091, −7.907101704595984497846658370590, −6.71969082001818199098112919408, −6.48899115552962724822888454894, −5.18276844955230411128101277011, −4.23136859617987607515088462925, −3.16384782959384238015569817843, −1.90118819029745969572108162732, −1.14921460066031805085775916728,
1.31380741600684764725411082491, 2.78477531359796432839210405784, 3.66311435835420917794965766917, 4.32651449728783781025388179014, 5.38388233176743433062316770880, 6.45256265743897919718379732435, 7.06346881896538749643305703881, 8.181438582043577841621021413402, 9.034962477470248660635723761695, 9.608505698958108577210564248840