L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − 3i·5-s + (−0.866 + 0.499i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (1 + 1.73i)9-s + (1.5 − 2.59i)10-s + (5.19 + 3i)11-s − 0.999·12-s + 0.999·14-s + (2.59 + 1.5i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + 2i·18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s − 1.34i·5-s + (−0.353 + 0.204i)6-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s + (1.56 + 0.904i)11-s − 0.288·12-s + 0.267·14-s + (0.670 + 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + 0.471i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78073 + 0.551653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78073 + 0.551653i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-5.19 + 3i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.1 + 7i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.59 - 1.5i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 + 3i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \) |
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
−11.82886036140134636675036524714, −10.90404238558420770391334321437, −9.559964482168443674386348551356, −8.975343567596513651595160161131, −7.72437897160471159367465389417, −6.77929654050765462482870108383, −5.33892699319055214569349783933, −4.69734735241645481312565641433, −3.93672797231837237755924166784, −1.66997696352796382074964464652,
1.55653915841502414727331083642, 3.18528608724977792782108047105, 4.06627112795307223189785650988, 5.85189780807458816715704525489, 6.49148667943659092833895167288, 7.22580839930106836723591290574, 8.697849266539438259622865848175, 9.850147166656173589126068592679, 10.84531898189319695833100174547, 11.62011912322396610554581162090