L(s) = 1 | + 8.47·2-s + (11.5 + 10.4i)3-s + 39.8·4-s − 35.7i·5-s + (98.3 + 88.3i)6-s − 11.5i·7-s + 66.9·8-s + (25.9 + 241. i)9-s − 302. i·10-s + (−202. + 346. i)11-s + (462. + 415. i)12-s − 933. i·13-s − 98.0i·14-s + (371. − 414. i)15-s − 708.·16-s − 25.0·17-s + ⋯ |
L(s) = 1 | + 1.49·2-s + (0.743 + 0.668i)3-s + 1.24·4-s − 0.638i·5-s + (1.11 + 1.00i)6-s − 0.0892i·7-s + 0.369·8-s + (0.106 + 0.994i)9-s − 0.957i·10-s + (−0.505 + 0.862i)11-s + (0.927 + 0.833i)12-s − 1.53i·13-s − 0.133i·14-s + (0.426 − 0.475i)15-s − 0.692·16-s − 0.0210·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.66851 + 0.571388i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.66851 + 0.571388i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-11.5 - 10.4i)T \) |
| 11 | \( 1 + (202. - 346. i)T \) |
good | 2 | \( 1 - 8.47T + 32T^{2} \) |
| 5 | \( 1 + 35.7iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 11.5iT - 1.68e4T^{2} \) |
| 13 | \( 1 + 933. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 25.0T + 1.41e6T^{2} \) |
| 19 | \( 1 + 743. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 594. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.77e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.40e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.23e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.03e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.64e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.93e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 7.24e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.69e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 3.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.05e5T + 8.58e9T^{2} \) |
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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
−15.30098077045425292172443314767, −14.64697943155063364887406951477, −13.19827844722227402159526455345, −12.72285364355068766988975842112, −10.91201365185255748112576176672, −9.386041376315941467985273451917, −7.72302421343714921652172577235, −5.45100230802851041695428517323, −4.38451809607134296854856953657, −2.80871458074465628853742241098,
2.46700860963799403565050440010, 3.88573204932314259153819131485, 5.98248101187934986595930523458, 7.19643992818012241844277503536, 9.003524721172329102120785171216, 11.14799908513775198771639206131, 12.32900847048789034994823791986, 13.47546510467402095957766588375, 14.22632439654475423919925983521, 14.99930908198336919723968286981