L(s) = 1 | + (−1.89 + 0.629i)2-s + (3.20 − 2.39i)4-s + (−5.84 − 3.37i)5-s + (−3.50 + 2.02i)7-s + (−4.58 + 6.55i)8-s + (13.2 + 2.72i)10-s + (4.13 + 7.16i)11-s + (7.66 + 4.42i)13-s + (5.38 − 6.05i)14-s + (4.57 − 15.3i)16-s + 28.6·17-s − 7.93·19-s + (−26.8 + 3.14i)20-s + (−12.3 − 11.0i)22-s + (25.3 + 14.6i)23-s + ⋯ |
L(s) = 1 | + (−0.949 + 0.314i)2-s + (0.801 − 0.597i)4-s + (−1.16 − 0.675i)5-s + (−0.501 + 0.289i)7-s + (−0.572 + 0.819i)8-s + (1.32 + 0.272i)10-s + (0.376 + 0.651i)11-s + (0.589 + 0.340i)13-s + (0.384 − 0.432i)14-s + (0.285 − 0.958i)16-s + 1.68·17-s − 0.417·19-s + (−1.34 + 0.157i)20-s + (−0.562 − 0.500i)22-s + (1.10 + 0.635i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.654757 + 0.333787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.654757 + 0.333787i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.89 - 0.629i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (5.84 + 3.37i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3.50 - 2.02i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.13 - 7.16i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.66 - 4.42i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 28.6T + 289T^{2} \) |
| 19 | \( 1 + 7.93T + 361T^{2} \) |
| 23 | \( 1 + (-25.3 - 14.6i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (15.7 - 9.08i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-40.8 - 23.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 13.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (31.0 - 53.7i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-26.5 - 46.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-12.8 + 7.43i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-11.4 + 19.9i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (51.3 - 29.6i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 17.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 54.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-62.9 + 36.3i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (4.34 + 7.53i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 28.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (0.200 + 0.348i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.01315071076787249700117884130, −11.33612328732875177236584891960, −10.05459023648176166706301893486, −9.174787779296375570900971873784, −8.272153864636004397610260412932, −7.43833954106956277222640748311, −6.32517536853320609645802428595, −4.93096186567404604738163797851, −3.34278501399268079000358852210, −1.17133042050787793592835165376,
0.68967144607779312716485425353, 3.04487185527646729622787773212, 3.81623089483409879945778819743, 6.09097922490986905245137620046, 7.17453726623012549050265823787, 7.967933778564543068154280194877, 8.915904328351308825028372004308, 10.19284507213943746263357183472, 10.86468433543463474284224704168, 11.74299869832734982830215294312