L(s) = 1 | + (0.161 + 1.40i)2-s + (−0.734 + 0.734i)3-s + (−1.94 + 0.452i)4-s + (1.90 + 1.17i)5-s + (−1.14 − 0.913i)6-s − 1.71·7-s + (−0.949 − 2.66i)8-s + 1.92i·9-s + (−1.34 + 2.86i)10-s + (2.82 − 2.82i)11-s + (1.09 − 1.76i)12-s + (2.59 − 2.59i)13-s + (−0.276 − 2.40i)14-s + (−2.25 + 0.537i)15-s + (3.59 − 1.76i)16-s − 1.89i·17-s + ⋯ |
L(s) = 1 | + (0.113 + 0.993i)2-s + (−0.423 + 0.423i)3-s + (−0.974 + 0.226i)4-s + (0.851 + 0.524i)5-s + (−0.469 − 0.372i)6-s − 0.648·7-s + (−0.335 − 0.941i)8-s + 0.640i·9-s + (−0.423 + 0.905i)10-s + (0.852 − 0.852i)11-s + (0.317 − 0.508i)12-s + (0.719 − 0.719i)13-s + (−0.0738 − 0.643i)14-s + (−0.583 + 0.138i)15-s + (0.897 − 0.440i)16-s − 0.460i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.530437 + 0.717422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.530437 + 0.717422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.161 - 1.40i)T \) |
| 5 | \( 1 + (-1.90 - 1.17i)T \) |
good | 3 | \( 1 + (0.734 - 0.734i)T - 3iT^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + (-2.82 + 2.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.59 + 2.59i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.89iT - 17T^{2} \) |
| 19 | \( 1 + (-2.89 - 2.89i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.00T + 23T^{2} \) |
| 29 | \( 1 + (6.72 + 6.72i)T + 29iT^{2} \) |
| 31 | \( 1 + 7.11T + 31T^{2} \) |
| 37 | \( 1 + (2.25 + 2.25i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.59iT - 41T^{2} \) |
| 43 | \( 1 + (-8.06 - 8.06i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.43iT - 47T^{2} \) |
| 53 | \( 1 + (0.481 + 0.481i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.08 + 3.08i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.46 - 3.46i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.80 - 1.80i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.379iT - 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + (-8.24 + 8.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + 6.50iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.68996239064655222808487895314, −13.77530817310429492927812657564, −13.03972804044890383851471878062, −11.30805685037572530670164721492, −10.08714537957999266014900457315, −9.174811255509286124549453427812, −7.65801959472136938697122119971, −6.17112246008388854500313263856, −5.55625292307375161099902779133, −3.62287289909574280848490993344,
1.59597343247988219945523923530, 3.84241418047159126184184669106, 5.55646904468619983149520103995, 6.77567862482055047861243959565, 9.098538138575268777919153573982, 9.463310613970549339634390998781, 10.92820682039079306552621468435, 12.14588760635826544622009928802, 12.74284993967291224400575811679, 13.71021569203682379967406508084