L(s) = 1 | + (0.965 + 0.258i)2-s + (−1.22 − 1.22i)3-s + (0.866 + 0.499i)4-s + (−0.866 − 1.49i)6-s + (−1.22 + 4.57i)7-s + (0.707 + 0.707i)8-s + 2.99i·9-s + (3 − 1.73i)11-s + (−0.448 − 1.67i)12-s + (1.22 + 4.57i)13-s + (−2.36 + 4.09i)14-s + (0.500 + 0.866i)16-s + (−0.776 + 2.89i)18-s − 3.19i·19-s + (7.09 − 4.09i)21-s + (3.34 − 0.896i)22-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.707 − 0.707i)3-s + (0.433 + 0.249i)4-s + (−0.353 − 0.612i)6-s + (−0.462 + 1.72i)7-s + (0.249 + 0.249i)8-s + 0.999i·9-s + (0.904 − 0.522i)11-s + (−0.129 − 0.482i)12-s + (0.339 + 1.26i)13-s + (−0.632 + 1.09i)14-s + (0.125 + 0.216i)16-s + (−0.183 + 0.683i)18-s − 0.733i·19-s + (1.54 − 0.894i)21-s + (0.713 − 0.191i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44767 + 0.689713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44767 + 0.689713i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.22 - 4.57i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 4.57i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 3.19iT - 19T^{2} \) |
| 23 | \( 1 + (-2.12 + 0.568i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-5.36 - 9.29i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.0980 - 0.169i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.79 + 5.79i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.448 + 0.120i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (5.79 + 1.55i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.79 - 5.79i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.76 - 4.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.34 - 1.43i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.26iT - 71T^{2} \) |
| 73 | \( 1 + (-3.67 + 3.67i)T - 73iT^{2} \) |
| 79 | \( 1 + (-8.66 + 5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.45 + 16.6i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 8.66T + 89T^{2} \) |
| 97 | \( 1 + (0.688 - 2.56i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.64699769152722848917927283745, −10.71285931123395591342840929599, −9.061986576647071942994340769514, −8.673975681971271637514635847904, −7.04496135548552481192316006040, −6.45280262606502187805465258822, −5.67957352155413737711685853154, −4.72545298102791268691171201093, −3.11740051329810810002918187879, −1.82068379500162274055569376777,
0.956360266069700077826968470920, 3.39523423587244600780480410898, 4.04807626796350292136293852205, 5.00720936963064245911273534203, 6.24682569448523036521458959149, 6.88630697482404772715717892107, 8.098327423991145272805029027983, 9.781534891779781550751103565739, 10.12951731416367708878342229742, 10.92655425604616279744589025050