| L(s) = 1 | + (1.97 − 0.332i)2-s + (−2.99 + 0.197i)3-s + (3.77 − 1.31i)4-s + (3.98 − 1.45i)5-s + (−5.83 + 1.38i)6-s + (2.54 − 0.448i)7-s + (7.01 − 3.84i)8-s + (8.92 − 1.18i)9-s + (7.37 − 4.18i)10-s + (0.818 − 2.24i)11-s + (−11.0 + 4.67i)12-s + (−3.62 − 3.04i)13-s + (4.87 − 1.73i)14-s + (−11.6 + 5.12i)15-s + (12.5 − 9.92i)16-s + (−8.77 + 15.2i)17-s + ⋯ |
| L(s) = 1 | + (0.986 − 0.166i)2-s + (−0.997 + 0.0657i)3-s + (0.944 − 0.328i)4-s + (0.796 − 0.290i)5-s + (−0.972 + 0.230i)6-s + (0.363 − 0.0641i)7-s + (0.876 − 0.480i)8-s + (0.991 − 0.131i)9-s + (0.737 − 0.418i)10-s + (0.0744 − 0.204i)11-s + (−0.920 + 0.389i)12-s + (−0.279 − 0.234i)13-s + (0.347 − 0.123i)14-s + (−0.776 + 0.341i)15-s + (0.784 − 0.620i)16-s + (−0.516 + 0.894i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.00281 - 0.466046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00281 - 0.466046i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.97 + 0.332i)T \) |
| 3 | \( 1 + (2.99 - 0.197i)T \) |
| good | 5 | \( 1 + (-3.98 + 1.45i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 0.448i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.818 + 2.24i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (3.62 + 3.04i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (8.77 - 15.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (13.3 - 7.72i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-13.3 - 2.35i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (33.3 - 27.9i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (46.8 + 8.25i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-23.0 + 39.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-59.1 - 49.5i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-11.2 + 30.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-36.6 + 6.45i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 35.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.4 - 86.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (17.4 + 98.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (18.1 - 21.6i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-40.2 - 23.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (18.0 + 31.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-58.8 - 70.1i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (50.7 + 60.4i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (31.6 + 54.8i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (74.9 + 27.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.93730383425321730484699439824, −12.73791918706083768718819819955, −11.22387551772606571736556102421, −10.70702337312369175629320821896, −9.399199610878691299626100725866, −7.42436426727561957567747501744, −6.09654679733759240928761716238, −5.37611304787907456649960456032, −4.07396381600464075666044365248, −1.75608193593333946879867248383,
2.17027230282178869346622642335, 4.38286901821088171222780227129, 5.47723147867387919221267571103, 6.48949163031182793202915937954, 7.47029160786638413879100381951, 9.497249750404448225771003697869, 10.82893203525068938814677501796, 11.46940323212215099868764294605, 12.63221257591525706494479260487, 13.41795149902790119127879492199
