L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.133 − 2.23i)5-s − 0.999·6-s + (1.73 + 2i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.23 + 1.86i)10-s + (2.5 − 4.33i)11-s + (0.866 − 0.499i)12-s − i·13-s + (−2.5 − 0.866i)14-s + (1 − 1.99i)15-s + (−0.5 − 0.866i)16-s + (1.73 + i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.0599 − 0.998i)5-s − 0.408·6-s + (0.654 + 0.755i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.389 + 0.590i)10-s + (0.753 − 1.30i)11-s + (0.249 − 0.144i)12-s − 0.277i·13-s + (−0.668 − 0.231i)14-s + (0.258 − 0.516i)15-s + (−0.125 − 0.216i)16-s + (0.420 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14118 + 0.106840i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14118 + 0.106840i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.133 + 2.23i)T \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + (11.2 - 6.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.19 + 3i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−12.14462747383253310133999046703, −11.53274759513886522619930169779, −10.14480761795054286053445159542, −9.223592974105919773077097800917, −8.359264956592400458377634175427, −7.940106893322321348302467614680, −6.05064931187087773186773629645, −5.20261531446482925378923494495, −3.59722019306561726796420886424, −1.53774337077871594567512854657,
1.73328792566260583794607486458, 3.20795553169858875821681464774, 4.57687865075359049495786459565, 6.83579701117063113614557987644, 7.19436015372326548269831519251, 8.313293496572333223869127736661, 9.584904743817233895065080360825, 10.24960616383089923338695279341, 11.39117332816808950014467798564, 12.02235753705840457009937498452