L(s) = 1 | + (−0.866 − 0.5i)2-s + 2.29·3-s + (0.499 + 0.866i)4-s + (−0.781 + 0.450i)5-s + (−1.98 − 1.14i)6-s − 0.999i·8-s + 2.24·9-s + 0.901·10-s − 4.33i·11-s + (1.14 + 1.98i)12-s + (−0.426 − 3.58i)13-s + (−1.78 + 1.03i)15-s + (−0.5 + 0.866i)16-s + (2.53 + 4.38i)17-s + (−1.94 − 1.12i)18-s − 6.17i·19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + 1.32·3-s + (0.249 + 0.433i)4-s + (−0.349 + 0.201i)5-s + (−0.809 − 0.467i)6-s − 0.353i·8-s + 0.749·9-s + 0.285·10-s − 1.30i·11-s + (0.330 + 0.572i)12-s + (−0.118 − 0.992i)13-s + (−0.461 + 0.266i)15-s + (−0.125 + 0.216i)16-s + (0.614 + 1.06i)17-s + (−0.458 − 0.264i)18-s − 1.41i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.695989445\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.695989445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (0.426 + 3.58i)T \) |
good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 + (0.781 - 0.450i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 4.33iT - 11T^{2} \) |
| 17 | \( 1 + (-2.53 - 4.38i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 6.17iT - 19T^{2} \) |
| 23 | \( 1 + (-4.22 + 7.31i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.09 - 1.89i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.756 - 0.436i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.124 + 0.0721i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.46 - 1.99i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.85 + 6.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.52 + 1.46i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.848 - 1.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.40 + 4.27i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + (2.83 + 1.63i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.466 - 0.269i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.26 + 5.65i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2iT - 83T^{2} \) |
| 89 | \( 1 + (-6.74 - 3.89i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 5.85i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.225826757730531790498504979410, −8.649318368000947415556903452859, −8.174447277499477491223494110206, −7.42471822307801487946286735515, −6.43542072733965545762923254625, −5.22045573082670020339398438337, −3.75665048126770450889713488829, −3.16153377788784082727715644930, −2.39442895795050530279497555609, −0.77022017054151044794088676821,
1.52127210973171362426092395785, 2.50084476597236616292336032379, 3.68423309757278616640309759146, 4.60682258143533295937524425684, 5.75159835046550881141229245482, 7.08491360256459071031073988456, 7.49268730444772458077462625987, 8.219731825755539249263644053488, 9.025779364930057911695737868218, 9.716811111984261824487635401435