L(s) = 1 | + (−0.866 + 1.11i)2-s + (1.58 + 0.707i)3-s + (−0.500 − 1.93i)4-s − 1.41i·5-s + (−2.15 + 1.15i)6-s + (1 + 2.44i)7-s + (2.59 + 1.11i)8-s + (2.00 + 2.23i)9-s + (1.58 + 1.22i)10-s + 3.46·11-s + (0.578 − 3.41i)12-s − 3.16·13-s + (−3.60 − 1.00i)14-s + (1.00 − 2.23i)15-s + (−3.5 + 1.93i)16-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.790i)2-s + (0.912 + 0.408i)3-s + (−0.250 − 0.968i)4-s − 0.632i·5-s + (−0.881 + 0.471i)6-s + (0.377 + 0.925i)7-s + (0.918 + 0.395i)8-s + (0.666 + 0.745i)9-s + (0.500 + 0.387i)10-s + 1.04·11-s + (0.167 − 0.985i)12-s − 0.877·13-s + (−0.963 − 0.268i)14-s + (0.258 − 0.577i)15-s + (−0.875 + 0.484i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.970565 + 0.616285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.970565 + 0.616285i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 1.11i)T \) |
| 3 | \( 1 + (-1.58 - 0.707i)T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.74iT - 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 9.89iT - 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 8.94iT - 71T^{2} \) |
| 73 | \( 1 + 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−13.17423131273702908563885848451, −12.02943461156794074059062774677, −10.62877994030827485317112776643, −9.441231182780008614042031486341, −8.894925011350854686043690034764, −8.128910159450559244278434742155, −6.88119832015917161821750676651, −5.39142267679829561118612244354, −4.32076100637348721087153216491, −2.05455184383410879499229456875,
1.63677850635857086974171701745, 3.18807791239472889795321765492, 4.27271080404863649582117291601, 6.89580469388421570873766067930, 7.47190556491913243450822279492, 8.653009923011391321683033146014, 9.612085067698435790213968570182, 10.52138067651388622433446232261, 11.55133274145353068797730683659, 12.54776503428423819263768222921