| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 8.36i·5-s − 2.82·8-s − 11.8i·10-s − 6·11-s − 17.8i·13-s + 4.00·16-s + 18.7i·17-s + 17.0i·19-s + 16.7i·20-s + 8.48·22-s − 13.4·23-s − 44.9·25-s + 25.2i·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 1.67i·5-s − 0.353·8-s − 1.18i·10-s − 0.545·11-s − 1.37i·13-s + 0.250·16-s + 1.10i·17-s + 0.895i·19-s + 0.836i·20-s + 0.385·22-s − 0.585·23-s − 1.79·25-s + 0.971i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1524896625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1524896625\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8.36iT - 25T^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 + 17.8iT - 169T^{2} \) |
| 17 | \( 1 - 18.7iT - 289T^{2} \) |
| 19 | \( 1 - 17.0iT - 361T^{2} \) |
| 23 | \( 1 + 13.4T + 529T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 - 14.7iT - 961T^{2} \) |
| 37 | \( 1 - 5.97T + 1.36e3T^{2} \) |
| 41 | \( 1 + 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 34.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 27.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 117. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 88.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 75.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 20.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 30.5iT - 9.40e3T^{2} \) |
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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
−10.43919152586400983996447821070, −9.982360924823034689165411037348, −8.672771076053353076753325318092, −7.75931696866241380524984806729, −7.33061103430535377171560288793, −6.17578822789067103604565838499, −5.64150141299988136493282801900, −3.77247317859307510834462998771, −2.98138068712985321195794434446, −1.90602529453517399240091474974,
0.06373964186619437688015810750, 1.28872787804274720326488253844, 2.45748055188312596414284691610, 4.16533571095818743581149091183, 4.94129551597148886018286601703, 5.88999274050024501795214163125, 7.09203064107951781326828318087, 7.87368999257230395151388903428, 8.815404039361831568740104405877, 9.270383344276213285888549544420
