L(s) = 1 | + (−0.654 + 0.755i)3-s + (1.50 − 1.18i)7-s + (−0.142 − 0.989i)9-s + (−0.911 − 1.28i)13-s + (1.02 + 0.809i)19-s + (−0.0913 + 1.91i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−1.03 + 1.44i)31-s + (−0.415 + 0.719i)37-s + (1.56 + 0.149i)39-s + (1.91 + 0.560i)43-s + (0.632 − 2.60i)49-s + (−1.28 + 0.247i)57-s + (−1.74 + 0.899i)61-s + ⋯ |
L(s) = 1 | + (−0.654 + 0.755i)3-s + (1.50 − 1.18i)7-s + (−0.142 − 0.989i)9-s + (−0.911 − 1.28i)13-s + (1.02 + 0.809i)19-s + (−0.0913 + 1.91i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−1.03 + 1.44i)31-s + (−0.415 + 0.719i)37-s + (1.56 + 0.149i)39-s + (1.91 + 0.560i)43-s + (0.632 − 2.60i)49-s + (−1.28 + 0.247i)57-s + (−1.74 + 0.899i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8942552070\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8942552070\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.995 - 0.0950i)T \) |
good | 5 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-1.50 + 1.18i)T + (0.235 - 0.971i)T^{2} \) |
| 11 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 13 | \( 1 + (0.911 + 1.28i)T + (-0.327 + 0.945i)T^{2} \) |
| 17 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 0.809i)T + (0.235 + 0.971i)T^{2} \) |
| 23 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1.03 - 1.44i)T + (-0.327 - 0.945i)T^{2} \) |
| 37 | \( 1 + (0.415 - 0.719i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 43 | \( 1 + (-1.91 - 0.560i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (1.74 - 0.899i)T + (0.580 - 0.814i)T^{2} \) |
| 71 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 73 | \( 1 + (0.0845 - 0.0436i)T + (0.580 - 0.814i)T^{2} \) |
| 79 | \( 1 + (1.15 - 0.110i)T + (0.981 - 0.189i)T^{2} \) |
| 83 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 89 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.48222591723940887044106924564, −9.981752955820438891302371204724, −8.762647907406122237621686611748, −7.76666410424777109602638297536, −7.19283717030113278322552253504, −5.77336609641005817828312978451, −4.99918833047044954782604113369, −4.33403961792326791213737396882, −3.16202895793486284863130512860, −1.19294425214675253123173380350,
1.67114332973320113597179253822, 2.49199561818217430056286652253, 4.50382773265263387192812890435, 5.24020138275956535097984404342, 5.94263530866907313120775555137, 7.28901739884144450511137937886, 7.60734406663180611501592426071, 8.866323870319026855278764441926, 9.377795611055026062765779338247, 10.94008290604613058494336588060