| L(s) = 1 | + (−0.721 − 2.91i)3-s + 2.23i·5-s − 1.03·7-s + (−7.95 + 4.20i)9-s + 19.5i·11-s + 16.3·13-s + (6.51 − 1.61i)15-s − 27.1i·17-s − 24.3·19-s + (0.747 + 3.01i)21-s − 19.9i·23-s − 5.00·25-s + (17.9 + 20.1i)27-s − 50.0i·29-s − 32.9·31-s + ⋯ |
| L(s) = 1 | + (−0.240 − 0.970i)3-s + 0.447i·5-s − 0.148·7-s + (−0.884 + 0.466i)9-s + 1.77i·11-s + 1.25·13-s + (0.434 − 0.107i)15-s − 1.59i·17-s − 1.28·19-s + (0.0356 + 0.143i)21-s − 0.866i·23-s − 0.200·25-s + (0.665 + 0.746i)27-s − 1.72i·29-s − 1.06·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5467638392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5467638392\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.721 + 2.91i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 1.03T + 49T^{2} \) |
| 11 | \( 1 - 19.5iT - 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 + 27.1iT - 289T^{2} \) |
| 19 | \( 1 + 24.3T + 361T^{2} \) |
| 23 | \( 1 + 19.9iT - 529T^{2} \) |
| 29 | \( 1 + 50.0iT - 841T^{2} \) |
| 31 | \( 1 + 32.9T + 961T^{2} \) |
| 37 | \( 1 - 0.942T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 68.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 14.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 34.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.54T + 4.48e3T^{2} \) |
| 71 | \( 1 - 10.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 14.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 58.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 37.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.7T + 9.40e3T^{2} \) |
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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
−9.450780152050111589610345082977, −8.478696545589145522107831802173, −7.59708624596974335861972481101, −6.83253121646072905296234274853, −6.31408171186323369639995078562, −5.14218085250983111100477166428, −4.08251923324343467376095670267, −2.63359074635672316837216921606, −1.80142892463743801400321034588, −0.17911007498449418572844778100,
1.41540662859297831916389150186, 3.39825357206911189770371464924, 3.73892120612457716816294461486, 5.01312983870746412434724191885, 5.96890025194919320383002171428, 6.35493562979170663582553334734, 8.160741832275033054817562563814, 8.622359352458115074280267681583, 9.218466618980534296665776899409, 10.44126717464752385296636452052
