| L(s) = 1 | + (−0.217 + 0.0968i)2-s + (−1.50 + 0.856i)3-s + (−1.30 + 1.44i)4-s + (1.93 − 1.12i)5-s + (0.244 − 0.332i)6-s + (−1.58 + 2.73i)7-s + (0.290 − 0.892i)8-s + (1.53 − 2.57i)9-s + (−0.310 + 0.432i)10-s + (−3.55 + 1.58i)11-s + (0.720 − 3.28i)12-s + (−5.89 − 2.62i)13-s + (0.0786 − 0.748i)14-s + (−1.94 + 3.35i)15-s + (−0.382 − 3.64i)16-s + (−2.22 + 6.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.153 + 0.0684i)2-s + (−0.869 + 0.494i)3-s + (−0.650 + 0.722i)4-s + (0.863 − 0.504i)5-s + (0.0997 − 0.135i)6-s + (−0.597 + 1.03i)7-s + (0.102 − 0.315i)8-s + (0.510 − 0.859i)9-s + (−0.0983 + 0.136i)10-s + (−1.07 + 0.476i)11-s + (0.207 − 0.949i)12-s + (−1.63 − 0.728i)13-s + (0.0210 − 0.200i)14-s + (−0.501 + 0.865i)15-s + (−0.0957 − 0.910i)16-s + (−0.538 + 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0379134 + 0.380346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0379134 + 0.380346i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 - 0.856i)T \) |
| 5 | \( 1 + (-1.93 + 1.12i)T \) |
| good | 2 | \( 1 + (0.217 - 0.0968i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (1.58 - 2.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.55 - 1.58i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (5.89 + 2.62i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (2.22 - 6.83i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.490 - 1.50i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.172 + 1.64i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-2.56 - 0.544i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 0.360i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (0.974 - 0.707i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-5.00 - 2.22i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (5.96 - 10.3i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.01 - 1.27i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.201 - 0.621i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.228 - 0.101i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (0.279 - 0.124i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (3.64 - 0.775i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-0.0407 - 0.125i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.55 + 1.13i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.79 + 0.594i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.0920 - 0.102i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-11.1 - 8.08i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (7.27 + 1.54i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.68661977045958215406479858568, −12.13291505609626896779756933385, −10.40096014038310107247846846787, −9.859413295577377508739453315640, −8.976615425598859505009694829811, −7.890328528874516888039312707447, −6.34565160745219821072805898896, −5.34265745912592489643171014283, −4.52690562234848047900939537018, −2.68088700755582449212401838037,
0.35208072466205742126433678311, 2.39384758763461818534447864439, 4.70872597963532370109891305183, 5.48050689242980987117032364255, 6.75552753106664323400945079850, 7.37227060578188998714366310102, 9.216804596762079373582233876030, 10.10868360546040691322278029882, 10.55971403355866226387751425428, 11.64778492188209100799088288544
