L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 − 0.866i)10-s + (−1.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + i·17-s − 0.999i·20-s + (0.499 − 0.866i)25-s − 1.73·26-s + (0.866 + 1.5i)29-s + (−0.866 − 0.499i)32-s + (0.5 + 0.866i)34-s + 1.73i·37-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 − 0.866i)10-s + (−1.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + i·17-s − 0.999i·20-s + (0.499 − 0.866i)25-s − 1.73·26-s + (0.866 + 1.5i)29-s + (−0.866 − 0.499i)32-s + (0.5 + 0.866i)34-s + 1.73i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.996545349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.996545349\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.652296718854423198192364428344, −8.796296669779050106366435545177, −7.74891021902168217495423669631, −6.71960862007963364035735181838, −5.97924060921549831247525692325, −5.09482030385721969592910086986, −4.65344081520144472969315772992, −3.29767632926102987660041848069, −2.43109394178940768097480918369, −1.33298622868522307760368747257,
2.19635934830532508241228443941, 2.72277560087324283984708393983, 4.05065653436278199447280501935, 4.92498080651624038295448019293, 5.64464347314300908626297058920, 6.57532934662455015950951018342, 7.12912821880700978230009479616, 7.85274251888205119445411399133, 9.068301695837407338645673682603, 9.672569713723557349007118403917