| L(s) = 1 | + (0.550 − 1.92i)2-s + (−1.70 + 0.457i)3-s + (−3.39 − 2.11i)4-s + (−3.11 − 0.833i)5-s + (−0.0595 + 3.53i)6-s + (3.27 − 6.18i)7-s + (−5.93 + 5.36i)8-s + (−5.08 + 2.93i)9-s + (−3.31 + 5.52i)10-s + (−16.6 + 4.47i)11-s + (6.76 + 2.06i)12-s + (−4.48 − 4.48i)13-s + (−10.0 − 9.70i)14-s + 5.69·15-s + (7.04 + 14.3i)16-s + (7.15 + 4.13i)17-s + ⋯ |
| L(s) = 1 | + (0.275 − 0.961i)2-s + (−0.569 + 0.152i)3-s + (−0.848 − 0.528i)4-s + (−0.622 − 0.166i)5-s + (−0.00993 + 0.589i)6-s + (0.468 − 0.883i)7-s + (−0.741 + 0.670i)8-s + (−0.564 + 0.326i)9-s + (−0.331 + 0.552i)10-s + (−1.51 + 0.406i)11-s + (0.564 + 0.171i)12-s + (−0.345 − 0.345i)13-s + (−0.720 − 0.693i)14-s + 0.379·15-s + (0.440 + 0.897i)16-s + (0.421 + 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0772448 + 0.515969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0772448 + 0.515969i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.550 + 1.92i)T \) |
| 7 | \( 1 + (-3.27 + 6.18i)T \) |
| good | 3 | \( 1 + (1.70 - 0.457i)T + (7.79 - 4.5i)T^{2} \) |
| 5 | \( 1 + (3.11 + 0.833i)T + (21.6 + 12.5i)T^{2} \) |
| 11 | \( 1 + (16.6 - 4.47i)T + (104. - 60.5i)T^{2} \) |
| 13 | \( 1 + (4.48 + 4.48i)T + 169iT^{2} \) |
| 17 | \( 1 + (-7.15 - 4.13i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-9.71 + 36.2i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-31.8 + 18.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (10.7 - 10.7i)T - 841iT^{2} \) |
| 31 | \( 1 + (22.8 + 13.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (12.1 - 45.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 43.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (34.5 + 34.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (3.29 - 1.90i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (16.3 - 4.38i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-13.5 - 50.4i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-6.29 + 23.4i)T + (-3.22e3 - 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-6.40 - 23.8i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 19.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (27.1 - 47.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-3.71 - 6.44i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (99.9 + 99.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (18.7 + 32.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 181. iT - 9.40e3T^{2} \) |
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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
−12.78476734271506992861346608238, −11.52759695514490621962454423589, −10.91268643816671198007155658283, −10.11756807982713863543632629429, −8.546743645463941772111765682714, −7.38426232884458853106412374238, −5.28477523818688744651867986609, −4.62545547433776728357967355733, −2.83249013292065576673974570621, −0.36326389160786411539140843483,
3.31759036014682755715678214222, 5.25104521426136974926023766424, 5.78256927475467574810490839240, 7.43329811833402250313910727901, 8.180527991944977810194240014653, 9.433175226234495462367098579733, 11.10146412275777489925568626259, 12.03235627090611061345783741878, 12.85236153406689933224219789291, 14.23170841978915529728774313758
