| L(s) = 1 | + (56.6 + 32.7i)2-s + (−1.10e4 + 1.62e4i)3-s + (−1.28e5 − 2.23e5i)4-s + (−2.62e6 + 1.51e6i)5-s + (−1.16e6 + 5.57e5i)6-s + (−1.47e6 + 2.55e6i)7-s − 3.40e7i·8-s + (−1.41e8 − 3.60e8i)9-s − 1.98e8·10-s + (−1.04e9 − 6.05e8i)11-s + (5.06e9 + 3.82e8i)12-s + (8.64e9 + 1.49e10i)13-s + (−1.67e8 + 9.66e7i)14-s + (4.49e9 − 5.94e10i)15-s + (−3.26e10 + 5.66e10i)16-s − 1.25e11i·17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.0638i)2-s + (−0.563 + 0.825i)3-s + (−0.491 − 0.851i)4-s + (−1.34 + 0.775i)5-s + (−0.115 + 0.0553i)6-s + (−0.0366 + 0.0633i)7-s − 0.253i·8-s + (−0.364 − 0.931i)9-s − 0.198·10-s + (−0.444 − 0.256i)11-s + (0.980 + 0.0741i)12-s + (0.815 + 1.41i)13-s + (−0.00810 + 0.00467i)14-s + (0.116 − 1.54i)15-s + (−0.475 + 0.823i)16-s − 1.05i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.744985 - 0.122318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744985 - 0.122318i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.10e4 - 1.62e4i)T \) |
| good | 2 | \( 1 + (-56.6 - 32.7i)T + (1.31e5 + 2.27e5i)T^{2} \) |
| 5 | \( 1 + (2.62e6 - 1.51e6i)T + (1.90e12 - 3.30e12i)T^{2} \) |
| 7 | \( 1 + (1.47e6 - 2.55e6i)T + (-8.14e14 - 1.41e15i)T^{2} \) |
| 11 | \( 1 + (1.04e9 + 6.05e8i)T + (2.77e18 + 4.81e18i)T^{2} \) |
| 13 | \( 1 + (-8.64e9 - 1.49e10i)T + (-5.62e19 + 9.73e19i)T^{2} \) |
| 17 | \( 1 + 1.25e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 - 2.32e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-2.16e12 + 1.25e12i)T + (1.62e24 - 2.80e24i)T^{2} \) |
| 29 | \( 1 + (9.21e12 + 5.32e12i)T + (1.05e26 + 1.82e26i)T^{2} \) |
| 31 | \( 1 + (-9.13e12 - 1.58e13i)T + (-3.49e26 + 6.05e26i)T^{2} \) |
| 37 | \( 1 + 1.98e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + (-1.49e14 + 8.64e13i)T + (5.35e28 - 9.28e28i)T^{2} \) |
| 43 | \( 1 + (-1.54e14 + 2.66e14i)T + (-1.26e29 - 2.18e29i)T^{2} \) |
| 47 | \( 1 + (1.53e15 + 8.86e14i)T + (6.26e29 + 1.08e30i)T^{2} \) |
| 53 | \( 1 + 1.23e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (-5.11e15 + 2.95e15i)T + (3.75e31 - 6.49e31i)T^{2} \) |
| 61 | \( 1 + (2.41e15 - 4.17e15i)T + (-6.83e31 - 1.18e32i)T^{2} \) |
| 67 | \( 1 + (1.60e14 + 2.77e14i)T + (-3.70e32 + 6.41e32i)T^{2} \) |
| 71 | \( 1 + 3.72e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 1.13e17T + 3.46e33T^{2} \) |
| 79 | \( 1 + (-9.38e16 + 1.62e17i)T + (-7.18e33 - 1.24e34i)T^{2} \) |
| 83 | \( 1 + (-8.86e16 - 5.11e16i)T + (1.74e34 + 3.02e34i)T^{2} \) |
| 89 | \( 1 - 2.88e16iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-1.50e17 + 2.60e17i)T + (-2.88e35 - 5.00e35i)T^{2} \) |
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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
−16.20171326721347280239503047843, −15.34391262160504434370960565006, −14.08639338199926527384150019847, −11.64169774992560943911129881254, −10.71352543376813753573453233759, −9.060790406517728536682336732160, −6.74194726419203522310669050000, −4.92972736253347342638086842480, −3.58053887335520664889770349834, −0.45830469622939823684133287512,
0.834342656384125783937777969994, 3.46875015658627261162376781953, 5.15909383897497307623297805443, 7.57544242997490654569637710362, 8.380679000287394739451422197776, 11.21578759135664618088668348911, 12.51814208682826761055617193388, 13.16453705799847154987663046642, 15.62262286790233356732803563632, 16.92474909575324066034783295186
