| L(s) = 1 | + (−1.16 + 1.62i)2-s + (−1.89 + 2.32i)3-s + (−1.26 − 3.79i)4-s + (−1.06 − 3.99i)5-s + (−1.56 − 5.79i)6-s + (2.82 − 1.63i)7-s + (7.63 + 2.38i)8-s + (−1.84 − 8.80i)9-s + (7.72 + 2.93i)10-s + (8.94 + 2.39i)11-s + (11.2 + 4.23i)12-s + (−4.06 − 15.1i)13-s + (−0.658 + 6.49i)14-s + (11.3 + 5.06i)15-s + (−12.8 + 9.59i)16-s + 27.9i·17-s + ⋯ |
| L(s) = 1 | + (−0.584 + 0.811i)2-s + (−0.630 + 0.776i)3-s + (−0.316 − 0.948i)4-s + (−0.213 − 0.798i)5-s + (−0.260 − 0.965i)6-s + (0.403 − 0.233i)7-s + (0.954 + 0.298i)8-s + (−0.204 − 0.978i)9-s + (0.772 + 0.293i)10-s + (0.812 + 0.217i)11-s + (0.935 + 0.352i)12-s + (−0.312 − 1.16i)13-s + (−0.0470 + 0.464i)14-s + (0.754 + 0.337i)15-s + (−0.800 + 0.599i)16-s + 1.64i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.808397 + 0.0435174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.808397 + 0.0435174i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 - 1.62i)T \) |
| 3 | \( 1 + (1.89 - 2.32i)T \) |
| good | 5 | \( 1 + (1.06 + 3.99i)T + (-21.6 + 12.5i)T^{2} \) |
| 7 | \( 1 + (-2.82 + 1.63i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.94 - 2.39i)T + (104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (4.06 + 15.1i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 - 27.9iT - 289T^{2} \) |
| 19 | \( 1 + (-22.9 + 22.9i)T - 361iT^{2} \) |
| 23 | \( 1 + (2.53 - 4.39i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-10.5 + 39.4i)T + (-728. - 420.5i)T^{2} \) |
| 31 | \( 1 + (-14.7 + 25.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (22.0 + 22.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-12.6 + 21.9i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-0.823 + 3.07i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (0.0912 - 0.0526i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-1.60 + 1.60i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (5.79 + 21.6i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (26.5 + 7.11i)T + (3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-14.6 - 54.7i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 38.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 75.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-65.8 - 114. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-10.2 + 38.2i)T + (-5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 + 150.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (16.8 + 29.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.84913801020502739804290231534, −11.66966985940539216596596434029, −10.61469420907437534803547565733, −9.694215002704385298939763892613, −8.746741116129117352084245537500, −7.69350319579777805440808947147, −6.23606643299068608439524673826, −5.18399392655893751234079944743, −4.18691003833010364642128156056, −0.806971148834331117911555977984,
1.44906873511623419347074126227, 3.07586211293968262431679180943, 4.90430394104282329049323113144, 6.74529593182633993738428107249, 7.43143765124465780844277985136, 8.745886216112017671753581096965, 9.932275817471966027280174633760, 11.13240747728673882414742081408, 11.75415144832250718345743462477, 12.28611919382208079678631320428
