| L(s) = 1 | + 4.52i·2-s − 12.4·4-s − 1.26·5-s − 20.1i·8-s − 5.72i·10-s + 41.5i·11-s + 85.7i·13-s − 8.54·16-s + 77.7·17-s + 48.6i·19-s + 15.7·20-s − 188.·22-s − 90.9i·23-s − 123.·25-s − 387.·26-s + ⋯ |
| L(s) = 1 | + 1.59i·2-s − 1.55·4-s − 0.113·5-s − 0.890i·8-s − 0.181i·10-s + 1.14i·11-s + 1.82i·13-s − 0.133·16-s + 1.10·17-s + 0.587i·19-s + 0.176·20-s − 1.82·22-s − 0.824i·23-s − 0.987·25-s − 2.92·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9045550385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9045550385\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.52iT - 8T^{2} \) |
| 5 | \( 1 + 1.26T + 125T^{2} \) |
| 11 | \( 1 - 41.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 85.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 77.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 90.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 151. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 88.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 90.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 334. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 880.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.1iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 341. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 921. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 413.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 954.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 29.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3iT - 9.12e5T^{2} \) |
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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
−11.57338570662302151753333662758, −10.02247709910330524375745340026, −9.403175912541014724560637839311, −8.335862256783987464805779173364, −7.55552488628339324747328701254, −6.76534852448736310224064727350, −5.97468807403038224240909089622, −4.80239384151850640446575384786, −4.02306902609063975798512460884, −1.94708686818213920740037993911,
0.30318481579374790167520254007, 1.40593536985928911933323742898, 3.10640221023999586996285041371, 3.39069189117067065760146952800, 4.98096641765911498856288108261, 5.92046215549100461721717567206, 7.55997093984052788201735433319, 8.467268105657158133833371192251, 9.445854536798516418237781032274, 10.37929263594421387156230443591
