| L(s) = 1 | + (5.65 − 0.0122i)2-s + (11.9 − 20.6i)3-s + (31.9 − 0.138i)4-s + (−26.3 + 15.2i)5-s + (67.2 − 117. i)6-s + (−129. + 7.22i)7-s + (181. − 1.17i)8-s + (−163. − 283. i)9-s + (−148. + 86.4i)10-s + (633. + 365. i)11-s + (379. − 663. i)12-s + 555. i·13-s + (−732. + 42.4i)14-s + 726. i·15-s + (1.02e3 − 8.83i)16-s + (−1.12e3 − 647. i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.00215i)2-s + (0.765 − 1.32i)3-s + (0.999 − 0.00431i)4-s + (−0.471 + 0.272i)5-s + (0.763 − 1.32i)6-s + (−0.998 + 0.0557i)7-s + (0.999 − 0.00647i)8-s + (−0.673 − 1.16i)9-s + (−0.470 + 0.273i)10-s + (1.57 + 0.911i)11-s + (0.760 − 1.33i)12-s + 0.911i·13-s + (−0.998 + 0.0578i)14-s + 0.834i·15-s + (0.999 − 0.00863i)16-s + (−0.940 − 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.65114 - 1.21644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65114 - 1.21644i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 + 0.0122i)T \) |
| 7 | \( 1 + (129. - 7.22i)T \) |
| good | 3 | \( 1 + (-11.9 + 20.6i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (26.3 - 15.2i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-633. - 365. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 555. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (1.12e3 + 647. i)T + (7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (387. + 671. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (580. - 335. i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (156. - 270. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.55e3 + 4.42e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 4.98e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.27e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (7.56e3 + 1.30e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.52e4 - 2.63e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.85e4 + 3.21e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-444. + 256. i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.77e4 - 1.59e4i)T + (6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.97e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.45e4 - 8.37e3i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.82e4 + 2.78e4i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (6.72e4 - 3.88e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 9.48e4iT - 8.58e9T^{2} \) |
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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
−15.60298628066532194052297974660, −14.45453660573841025003118483323, −13.52583165971677081730491592652, −12.48040182972817466249200309725, −11.56584226920373854782597394460, −9.200249498373803566513008782698, −7.15184947826071727586949840289, −6.65695999598553749323688811406, −3.82093092759705637544247959057, −2.04118712677583641430310718836,
3.31774053365296784186618039729, 4.20613640040779328759439391841, 6.23231542929433718570806143484, 8.463601700365522340773257592229, 9.907322024375574374380048534369, 11.27153834500375116431290678017, 12.79774361717135814318815413613, 14.12073737111090241210156182292, 15.12382999524347650367962240017, 15.99865244224270121296387845782
