| L(s) = 1 | + 2-s − 1.78·3-s + 4-s + 0.747·5-s − 1.78·6-s + 2.14·7-s + 8-s + 0.194·9-s + 0.747·10-s + 2.79·11-s − 1.78·12-s + 5.04·13-s + 2.14·14-s − 1.33·15-s + 16-s + 17-s + 0.194·18-s − 1.63·19-s + 0.747·20-s − 3.83·21-s + 2.79·22-s − 2.29·23-s − 1.78·24-s − 4.44·25-s + 5.04·26-s + 5.01·27-s + 2.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.03·3-s + 0.5·4-s + 0.334·5-s − 0.729·6-s + 0.810·7-s + 0.353·8-s + 0.0647·9-s + 0.236·10-s + 0.841·11-s − 0.515·12-s + 1.40·13-s + 0.572·14-s − 0.345·15-s + 0.250·16-s + 0.242·17-s + 0.0458·18-s − 0.375·19-s + 0.167·20-s − 0.835·21-s + 0.594·22-s − 0.479·23-s − 0.364·24-s − 0.888·25-s + 0.990·26-s + 0.965·27-s + 0.405·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.449504551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.449504551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 - 0.747T + 5T^{2} \) |
| 7 | \( 1 - 2.14T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 - 5.81T + 53T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 - 0.255T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 3.86T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
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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
−9.140569790987475734332090869672, −8.309049642499603857356149723823, −7.44402525810224857738013502509, −6.24721004768438535046556269396, −6.08332897338592884653151502600, −5.25383201535329929092182007910, −4.33042371906970624396420751689, −3.59570201888751246528528719646, −2.11725860682135407810277607512, −1.06604073233849546999920157907,
1.06604073233849546999920157907, 2.11725860682135407810277607512, 3.59570201888751246528528719646, 4.33042371906970624396420751689, 5.25383201535329929092182007910, 6.08332897338592884653151502600, 6.24721004768438535046556269396, 7.44402525810224857738013502509, 8.309049642499603857356149723823, 9.140569790987475734332090869672
