| L(s) = 1 | + (1.39 − 0.245i)2-s + (−1.56 − 2.55i)3-s + (1.87 − 0.684i)4-s + (1.85 − 2.21i)5-s + (−2.81 − 3.17i)6-s + (2.17 + 0.791i)7-s + (2.44 − 1.41i)8-s + (−4.07 + 8.02i)9-s + (2.04 − 3.54i)10-s + (−0.401 − 0.478i)11-s + (−4.69 − 3.73i)12-s + (−4.06 + 23.0i)13-s + (3.22 + 0.568i)14-s + (−8.58 − 1.27i)15-s + (3.06 − 2.57i)16-s + (2.71 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (−0.523 − 0.852i)3-s + (0.469 − 0.171i)4-s + (0.371 − 0.443i)5-s + (−0.468 − 0.529i)6-s + (0.310 + 0.113i)7-s + (0.306 − 0.176i)8-s + (−0.452 + 0.891i)9-s + (0.204 − 0.354i)10-s + (−0.0364 − 0.0434i)11-s + (−0.391 − 0.311i)12-s + (−0.312 + 1.77i)13-s + (0.230 + 0.0405i)14-s + (−0.572 − 0.0851i)15-s + (0.191 − 0.160i)16-s + (0.159 + 0.0922i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.34354 - 0.656233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34354 - 0.656233i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.245i)T \) |
| 3 | \( 1 + (1.56 + 2.55i)T \) |
| good | 5 | \( 1 + (-1.85 + 2.21i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.17 - 0.791i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (0.401 + 0.478i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (4.06 - 23.0i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-2.71 - 1.56i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-2.04 - 3.53i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (15.5 + 42.6i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-19.6 + 3.46i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (42.6 - 15.5i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-18.7 + 32.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (45.6 + 8.05i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-46.3 + 38.9i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-1.06 + 2.91i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 79.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (41.9 - 50.0i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (33.1 + 12.0i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (4.92 - 27.9i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (70.7 + 40.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (12.6 + 21.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-11.5 - 65.5i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-49.1 + 8.66i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-111. + 64.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-107. + 90.5i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−14.50746877639781297391740731437, −13.79393555398054120187863889657, −12.57729168392786659695421338675, −11.87964851835524931595408677656, −10.67231295341103942776613561237, −8.883365622216489015368833170959, −7.21892607507845779132002803433, −6.00955182321320734971830951379, −4.63955033684825289650563767454, −1.94395402201650362130461327538,
3.27936556529763431089611201038, 5.02302622160369566363924687004, 6.07257726493750000589360452713, 7.80014655059487420909978670640, 9.755448906587822172692975534851, 10.70270539208830089317235173992, 11.78342261320214771345683433646, 13.07370372014809379021288880818, 14.38061859726868332974800811368, 15.22497596551513550842612500670
