L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.23 + 0.712i)5-s + (−0.484 + 2.60i)7-s + 0.999i·8-s + (0.712 + 1.23i)10-s + (−2.51 + 2.16i)11-s − 2.39·13-s + (−1.72 + 2.01i)14-s + (−0.5 + 0.866i)16-s + (−0.529 − 0.917i)17-s + (−2.37 + 4.11i)19-s + 1.42i·20-s + (−3.25 + 0.620i)22-s + (−1.72 + 2.97i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.551 + 0.318i)5-s + (−0.183 + 0.983i)7-s + 0.353i·8-s + (0.225 + 0.390i)10-s + (−0.757 + 0.653i)11-s − 0.663·13-s + (−0.459 + 0.537i)14-s + (−0.125 + 0.216i)16-s + (−0.128 − 0.222i)17-s + (−0.544 + 0.943i)19-s + 0.318i·20-s + (−0.694 + 0.132i)22-s + (−0.358 + 0.621i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.873722495\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.873722495\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.484 - 2.60i)T \) |
| 11 | \( 1 + (2.51 - 2.16i)T \) |
good | 5 | \( 1 + (-1.23 - 0.712i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 + (0.529 + 0.917i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.72 - 2.97i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.13iT - 29T^{2} \) |
| 31 | \( 1 + (-0.122 + 0.0706i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.19 + 2.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 - 5.73iT - 43T^{2} \) |
| 47 | \( 1 + (-3.08 - 1.78i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.98 - 3.44i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.71 + 2.72i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.58 - 9.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.567 + 0.983i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + (0.969 + 1.67i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.82 + 2.20i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.73T + 83T^{2} \) |
| 89 | \( 1 + (-1.67 - 0.969i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18.9iT - 97T^{2} \) |
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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
−9.848461695597534663680076413869, −9.179346972302464173891871506526, −8.034899499463601689648671821495, −7.49566899195100703859780923395, −6.33285750656630612303890363215, −5.84014480133604113112631772090, −5.00117421620659128844348133551, −4.02650199703434059261992250949, −2.66130404565838807629442968879, −2.14804023082218385170703661197,
0.57021288677834542031700492954, 2.05540506580530570790272859074, 3.07536597452055038577189657968, 4.15743246544754358150665662597, 4.96994079417732404393252716341, 5.78615880000393379272655664790, 6.71580242760573700060537676726, 7.49014145971858912679597398993, 8.515611593386479753874107587847, 9.429721783695118892279649802213