| L(s) = 1 | + 7-s + 4·11-s + 6·13-s − 6·17-s − 4·19-s − 8·23-s + 10·29-s + 4·31-s − 6·37-s − 6·41-s + 4·43-s − 12·47-s + 49-s − 6·53-s + 4·59-s + 2·61-s + 4·67-s + 2·73-s + 4·77-s − 8·79-s + 12·83-s − 14·89-s + 6·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1.85·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.234·73-s + 0.455·77-s − 0.900·79-s + 1.31·83-s − 1.48·89-s + 0.628·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.410583083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.410583083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.82963408805630, −13.37894157472142, −12.80419067705971, −12.12014813787649, −11.85616578101454, −11.25364195266785, −10.89399335935553, −10.37443063885393, −9.814546585172261, −9.235238716064472, −8.551030377082671, −8.369066982254790, −8.084264929509647, −6.879099597317495, −6.634309256132328, −6.305012507674410, −5.730290910640660, −4.870580186384838, −4.310562697021177, −4.004974146959983, −3.374127156282428, −2.573856686608177, −1.764866732608253, −1.452690112817287, −0.4862014861075686,
0.4862014861075686, 1.452690112817287, 1.764866732608253, 2.573856686608177, 3.374127156282428, 4.004974146959983, 4.310562697021177, 4.870580186384838, 5.730290910640660, 6.305012507674410, 6.634309256132328, 6.879099597317495, 8.084264929509647, 8.369066982254790, 8.551030377082671, 9.235238716064472, 9.814546585172261, 10.37443063885393, 10.89399335935553, 11.25364195266785, 11.85616578101454, 12.12014813787649, 12.80419067705971, 13.37894157472142, 13.82963408805630
