| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 2·13-s − 2·14-s + 16-s − 2·17-s − 2·23-s + 2·26-s + 2·28-s + 4·29-s − 4·31-s − 32-s + 2·34-s − 2·37-s − 4·41-s − 10·43-s + 2·46-s + 6·47-s − 3·49-s − 2·52-s + 6·53-s − 2·56-s − 4·58-s − 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.417·23-s + 0.392·26-s + 0.377·28-s + 0.742·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.624·41-s − 1.52·43-s + 0.294·46-s + 0.875·47-s − 3/7·49-s − 0.277·52-s + 0.824·53-s − 0.267·56-s − 0.525·58-s − 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8922351150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8922351150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.32619613765921, −12.66786614408281, −12.08067198493416, −11.91621516872211, −11.26521118653654, −10.86346851980869, −10.38670383495685, −9.939719798410237, −9.443170474937179, −8.831997997015379, −8.537844058998259, −7.918170953751480, −7.626523186858662, −6.875210634970109, −6.685024252871197, −5.885338742187953, −5.398760391625430, −4.788420135067354, −4.375121765030151, −3.604133187505817, −3.007556592926593, −2.322818401133953, −1.791912398176953, −1.259199273715296, −0.3094789064764301,
0.3094789064764301, 1.259199273715296, 1.791912398176953, 2.322818401133953, 3.007556592926593, 3.604133187505817, 4.375121765030151, 4.788420135067354, 5.398760391625430, 5.885338742187953, 6.685024252871197, 6.875210634970109, 7.626523186858662, 7.918170953751480, 8.537844058998259, 8.831997997015379, 9.443170474937179, 9.939719798410237, 10.38670383495685, 10.86346851980869, 11.26521118653654, 11.91621516872211, 12.08067198493416, 12.66786614408281, 13.32619613765921
