L(s) = 1 | + (−0.913 − 2.04i)5-s − 4.62i·7-s + (−4.00 − 2.90i)11-s + (2.21 + 3.04i)13-s + (−2.55 − 0.831i)17-s + (−1.81 + 5.58i)19-s + (−3.92 + 5.40i)23-s + (−3.33 + 3.72i)25-s + (0.370 + 1.14i)29-s + (1.02 − 3.14i)31-s + (−9.44 + 4.22i)35-s + (1.10 + 1.51i)37-s + (−2.45 + 1.78i)41-s − 10.6i·43-s + (−0.246 + 0.0801i)47-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.912i)5-s − 1.74i·7-s + (−1.20 − 0.877i)11-s + (0.613 + 0.844i)13-s + (−0.620 − 0.201i)17-s + (−0.416 + 1.28i)19-s + (−0.818 + 1.12i)23-s + (−0.666 + 0.745i)25-s + (0.0688 + 0.212i)29-s + (0.183 − 0.564i)31-s + (−1.59 + 0.714i)35-s + (0.181 + 0.249i)37-s + (−0.383 + 0.278i)41-s − 1.62i·43-s + (−0.0359 + 0.0116i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0289340 - 0.661249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0289340 - 0.661249i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.913 + 2.04i)T \) |
good | 7 | \( 1 + 4.62iT - 7T^{2} \) |
| 11 | \( 1 + (4.00 + 2.90i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.21 - 3.04i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.55 + 0.831i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.81 - 5.58i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.92 - 5.40i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.370 - 1.14i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 3.14i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.10 - 1.51i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.45 - 1.78i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (0.246 - 0.0801i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.31 + 3.02i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.78 + 5.65i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (5.07 + 3.68i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.43 + 0.791i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.68 + 8.25i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.86 + 3.94i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.85 + 11.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-8.45 - 2.74i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (11.7 + 8.56i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.79 - 1.23i)T + (78.4 - 57.0i)T^{2} \) |
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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
−9.823669414387530598994578379220, −8.696246928719496458114935082705, −8.001040728978132930789806631416, −7.36208560718308170280921098853, −6.23075018416196548614787374532, −5.19796194336476958559660629974, −4.12270023331486617692233482831, −3.61137794845300739729779871369, −1.66965784080054494359208096259, −0.30263402669370592692460915444,
2.45270900868249519709574349972, 2.71758843552440202818087334319, 4.31872312422232711846464853708, 5.36559661067127801474549029226, 6.20324984698411926258148355870, 7.07131984756742194330626904278, 8.182692178222500282750422260245, 8.621568070715753000227329301897, 9.779467840452060528417709816229, 10.57895013656718094432618241530