| L(s) = 1 | − 3-s − 4·7-s + 9-s + 11-s − 2·13-s − 6·17-s − 4·19-s + 4·21-s − 8·23-s − 27-s − 10·29-s − 33-s + 10·37-s + 2·39-s − 6·41-s − 8·43-s + 9·49-s + 6·51-s + 2·53-s + 4·57-s − 12·59-s − 2·61-s − 4·63-s − 12·67-s + 8·69-s − 8·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.85·29-s − 0.174·33-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s + 9/7·49-s + 0.840·51-s + 0.274·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.503·63-s − 1.46·67-s + 0.963·69-s − 0.949·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.85736137469615, −16.31620606222687, −15.76685873284434, −15.08947925518631, −14.83917504528131, −13.69660668234614, −13.41244406472001, −12.76039855460874, −12.41600058877774, −11.62686799929985, −11.21827275594741, −10.36053689636559, −10.00751782207997, −9.316361302628713, −8.917255673867525, −7.968417875534327, −7.287624164236929, −6.571411449369204, −6.225029529661103, −5.682931457420132, −4.617926002007811, −4.120841655776253, −3.388141190114969, −2.440905849008425, −1.719396231512600, 0, 0,
1.719396231512600, 2.440905849008425, 3.388141190114969, 4.120841655776253, 4.617926002007811, 5.682931457420132, 6.225029529661103, 6.571411449369204, 7.287624164236929, 7.968417875534327, 8.917255673867525, 9.316361302628713, 10.00751782207997, 10.36053689636559, 11.21827275594741, 11.62686799929985, 12.41600058877774, 12.76039855460874, 13.41244406472001, 13.69660668234614, 14.83917504528131, 15.08947925518631, 15.76685873284434, 16.31620606222687, 16.85736137469615
