L(s) = 1 | − 687. i·2-s + (1.00e5 − 1.89e4i)3-s + 1.62e6·4-s − 6.28e6·5-s + (−1.30e7 − 6.91e7i)6-s + (4.61e8 − 5.87e8i)7-s − 2.55e9i·8-s + (9.74e9 − 3.80e9i)9-s + 4.32e9i·10-s − 6.92e10i·11-s + (1.63e11 − 3.07e10i)12-s + 2.42e11i·13-s + (−4.04e11 − 3.17e11i)14-s + (−6.31e11 + 1.19e11i)15-s + 1.64e12·16-s − 6.97e12·17-s + ⋯ |
L(s) = 1 | − 0.474i·2-s + (0.982 − 0.185i)3-s + 0.774·4-s − 0.287·5-s + (−0.0879 − 0.466i)6-s + (0.617 − 0.786i)7-s − 0.842i·8-s + (0.931 − 0.364i)9-s + 0.136i·10-s − 0.804i·11-s + (0.761 − 0.143i)12-s + 0.486i·13-s + (−0.373 − 0.293i)14-s + (−0.282 + 0.0533i)15-s + 0.374·16-s − 0.838·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(3.801691670\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.801691670\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.00e5 + 1.89e4i)T \) |
| 7 | \( 1 + (-4.61e8 + 5.87e8i)T \) |
good | 2 | \( 1 + 687. iT - 2.09e6T^{2} \) |
| 5 | \( 1 + 6.28e6T + 4.76e14T^{2} \) |
| 11 | \( 1 + 6.92e10iT - 7.40e21T^{2} \) |
| 13 | \( 1 - 2.42e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 + 6.97e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.12e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 5.39e13iT - 3.94e28T^{2} \) |
| 29 | \( 1 - 2.80e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 2.80e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 3.19e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.94e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 9.38e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.19e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.47e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 5.35e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 9.25e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 8.77e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.46e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 + 4.34e19iT - 1.34e39T^{2} \) |
| 79 | \( 1 - 6.43e18T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.21e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 6.58e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.64e20iT - 5.27e41T^{2} \) |
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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
−13.05362331932650200850624780094, −11.54192229130089585140981207725, −10.60301188917178813514578624872, −9.015046749838678289862319598985, −7.67141010926132137178789407277, −6.67092450983964325321048547564, −4.32735183106917979206405347064, −3.14083044678403226985210162412, −1.93744942030031151154076949096, −0.788896054355541019855023217248,
1.75619376627039539980881913141, 2.60180704965426097858079436955, 4.27391834869052166392451016366, 5.88502181177408097157873999120, 7.53168331608007731618801184455, 8.228644282397628108812763494427, 9.744375912775645174113720543176, 11.28381269597124853464043030172, 12.55366178958002189946992462008, 14.21275202912165010471822473005