| L(s) = 1 | + (−0.978 − 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (0.279 + 0.310i)5-s + (−0.309 + 0.951i)6-s + (−2.03 + 1.69i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.208 − 0.361i)10-s + (2.44 − 2.24i)11-s + (0.5 − 0.866i)12-s + (−1.95 − 6.00i)13-s + (2.33 − 1.23i)14-s + (0.338 − 0.245i)15-s + (0.669 + 0.743i)16-s + (5.99 − 1.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.0603 − 0.574i)3-s + (0.456 + 0.203i)4-s + (0.125 + 0.138i)5-s + (−0.126 + 0.388i)6-s + (−0.767 + 0.641i)7-s + (−0.286 − 0.207i)8-s + (−0.326 − 0.0693i)9-s + (−0.0660 − 0.114i)10-s + (0.736 − 0.676i)11-s + (0.144 − 0.250i)12-s + (−0.541 − 1.66i)13-s + (0.625 − 0.330i)14-s + (0.0872 − 0.0634i)15-s + (0.167 + 0.185i)16-s + (1.45 − 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583570 - 0.654242i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583570 - 0.654242i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 7 | \( 1 + (2.03 - 1.69i)T \) |
| 11 | \( 1 + (-2.44 + 2.24i)T \) |
| good | 5 | \( 1 + (-0.279 - 0.310i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (1.95 + 6.00i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-5.99 + 1.27i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (3.10 - 1.38i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-4.39 + 7.61i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.21 - 2.33i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.39 - 1.54i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.0897 + 0.854i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (3.13 + 2.27i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.07T + 43T^{2} \) |
| 47 | \( 1 + (0.168 - 0.0748i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.28 + 4.75i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-4.08 - 1.82i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (4.56 + 5.07i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-5.77 - 10.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.960 - 2.95i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.63 + 1.61i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (13.1 + 2.78i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (3.63 - 11.1i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.82 + 8.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.35 - 7.26i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.57723850784674948984312015941, −10.00239852663144811874749180667, −8.861628702536316416833384167768, −8.269291095374753729193156802957, −7.18841798416602777478942968983, −6.26754902502443941116308625793, −5.43581588571268933536766256001, −3.38453426686751109362956086881, −2.54479776516078172802576334607, −0.69774208166189731822038223223,
1.62620854501590096215056414461, 3.39888658961389995784793217326, 4.43627824554930561012423540894, 5.76940692486940090355131221968, 6.92812025162701840311763424177, 7.48889247039049648994424733689, 8.994719590678502200803183314171, 9.501836135703507443333986698834, 10.04386992582725116941397904764, 11.18181318746146822686487723727
