L(s) = 1 | + (−1.99 − 0.178i)2-s + 5.14i·3-s + (3.93 + 0.709i)4-s + (3.67 + 3.39i)5-s + (0.915 − 10.2i)6-s + (−3.38 + 6.12i)7-s + (−7.71 − 2.11i)8-s − 17.4·9-s + (−6.71 − 7.41i)10-s + 18.5i·11-s + (−3.64 + 20.2i)12-s − 12.0i·13-s + (7.83 − 11.5i)14-s + (−17.4 + 18.8i)15-s + (14.9 + 5.58i)16-s + 19.5·17-s + ⋯ |
L(s) = 1 | + (−0.996 − 0.0890i)2-s + 1.71i·3-s + (0.984 + 0.177i)4-s + (0.734 + 0.678i)5-s + (0.152 − 1.70i)6-s + (−0.483 + 0.875i)7-s + (−0.964 − 0.264i)8-s − 1.93·9-s + (−0.671 − 0.741i)10-s + 1.68i·11-s + (−0.304 + 1.68i)12-s − 0.925i·13-s + (0.559 − 0.828i)14-s + (−1.16 + 1.25i)15-s + (0.937 + 0.349i)16-s + 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0255960 - 0.980982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0255960 - 0.980982i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.99 + 0.178i)T \) |
| 5 | \( 1 + (-3.67 - 3.39i)T \) |
| 7 | \( 1 + (3.38 - 6.12i)T \) |
good | 3 | \( 1 - 5.14iT - 9T^{2} \) |
| 11 | \( 1 - 18.5iT - 121T^{2} \) |
| 13 | \( 1 + 12.0iT - 169T^{2} \) |
| 17 | \( 1 - 19.5T + 289T^{2} \) |
| 19 | \( 1 - 6.87T + 361T^{2} \) |
| 23 | \( 1 + 20.3iT - 529T^{2} \) |
| 29 | \( 1 - 2.31iT - 841T^{2} \) |
| 31 | \( 1 - 6.49iT - 961T^{2} \) |
| 37 | \( 1 - 30.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 70.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 68.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 14.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 23.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 65.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 38.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 66.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 49.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 30.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.8T + 9.40e3T^{2} \) |
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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
−11.81676761770759872426969522490, −10.55090556363112533584841553304, −10.10763645119559956741749849726, −9.597935771522395041956003142344, −8.755045119455244415038776768386, −7.39226379521450579316986576491, −6.06379043963867598761439586290, −5.14948542587829976395380138041, −3.39605887466163086885588207516, −2.42031188808324236846418593646,
0.70144980415780932233649602561, 1.51493209001398993303545062050, 3.09470523941188419536008176429, 5.74857821047890179170219248892, 6.33971851650961431250841316768, 7.36663441267592629109089413566, 8.154409831156560508818901906454, 9.055663758901258913155790352480, 10.03828627799914501112439830186, 11.31872975164293268937095724710