L(s) = 1 | − 2·5-s − 7-s + 6·13-s − 2·17-s − 4·19-s − 2·23-s − 25-s + 8·29-s + 4·31-s + 2·35-s − 6·37-s + 10·41-s + 4·43-s + 4·47-s + 49-s + 4·53-s + 12·59-s + 2·61-s − 12·65-s + 12·67-s − 6·71-s + 2·73-s + 8·79-s + 4·85-s + 14·89-s − 6·91-s + 8·95-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s + 1.48·29-s + 0.718·31-s + 0.338·35-s − 0.986·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.549·53-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 1.46·67-s − 0.712·71-s + 0.234·73-s + 0.900·79-s + 0.433·85-s + 1.48·89-s − 0.628·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.854793873\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.854793873\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.27748472002773, −13.66724269605382, −13.39072161074875, −12.74390817543260, −12.19255771316987, −11.87480086598248, −11.10506963603037, −10.88659672739365, −10.31869430397847, −9.725305451152607, −8.964798762087992, −8.567230184827128, −8.179985721899496, −7.640513643522045, −6.809797816843382, −6.504213730742881, −5.947140170891275, −5.295765390739739, −4.420745666612079, −3.969903879268461, −3.670653078195369, −2.757899517090664, −2.194619735293886, −1.145336376926585, −0.5262358768563536,
0.5262358768563536, 1.145336376926585, 2.194619735293886, 2.757899517090664, 3.670653078195369, 3.969903879268461, 4.420745666612079, 5.295765390739739, 5.947140170891275, 6.504213730742881, 6.809797816843382, 7.640513643522045, 8.179985721899496, 8.567230184827128, 8.964798762087992, 9.725305451152607, 10.31869430397847, 10.88659672739365, 11.10506963603037, 11.87480086598248, 12.19255771316987, 12.74390817543260, 13.39072161074875, 13.66724269605382, 14.27748472002773