| L(s) = 1 | + 3-s + 6·7-s − 26·9-s + 19·11-s + 12·13-s + 75·17-s + 91·19-s + 6·21-s − 174·23-s − 53·27-s + 272·29-s − 230·31-s + 19·33-s − 182·37-s + 12·39-s + 117·41-s + 372·43-s + 52·47-s − 307·49-s + 75·51-s − 402·53-s + 91·57-s − 312·59-s − 170·61-s − 156·63-s + 763·67-s − 174·69-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 0.323·7-s − 0.962·9-s + 0.520·11-s + 0.256·13-s + 1.07·17-s + 1.09·19-s + 0.0623·21-s − 1.57·23-s − 0.377·27-s + 1.74·29-s − 1.33·31-s + 0.100·33-s − 0.808·37-s + 0.0492·39-s + 0.445·41-s + 1.31·43-s + 0.161·47-s − 0.895·49-s + 0.205·51-s − 1.04·53-s + 0.211·57-s − 0.688·59-s − 0.356·61-s − 0.311·63-s + 1.39·67-s − 0.303·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.345628517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345628517\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 19 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 75 T + p^{3} T^{2} \) |
| 19 | \( 1 - 91 T + p^{3} T^{2} \) |
| 23 | \( 1 + 174 T + p^{3} T^{2} \) |
| 29 | \( 1 - 272 T + p^{3} T^{2} \) |
| 31 | \( 1 + 230 T + p^{3} T^{2} \) |
| 37 | \( 1 + 182 T + p^{3} T^{2} \) |
| 41 | \( 1 - 117 T + p^{3} T^{2} \) |
| 43 | \( 1 - 372 T + p^{3} T^{2} \) |
| 47 | \( 1 - 52 T + p^{3} T^{2} \) |
| 53 | \( 1 + 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 312 T + p^{3} T^{2} \) |
| 61 | \( 1 + 170 T + p^{3} T^{2} \) |
| 67 | \( 1 - 763 T + p^{3} T^{2} \) |
| 71 | \( 1 + 52 T + p^{3} T^{2} \) |
| 73 | \( 1 - 981 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1054 T + p^{3} T^{2} \) |
| 83 | \( 1 - 351 T + p^{3} T^{2} \) |
| 89 | \( 1 - 799 T + p^{3} T^{2} \) |
| 97 | \( 1 + 962 T + p^{3} T^{2} \) |
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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
−9.086973153737171919351246312674, −8.107355639526554076519300864650, −7.73092106136383601890635755188, −6.53178158737576650858416159924, −5.77181002984230338512433826895, −5.01684186943483192609531854215, −3.81798844213093599147925979639, −3.09031329764211998538803281526, −1.90673195090298212021105079156, −0.74104423450633238584822198309,
0.74104423450633238584822198309, 1.90673195090298212021105079156, 3.09031329764211998538803281526, 3.81798844213093599147925979639, 5.01684186943483192609531854215, 5.77181002984230338512433826895, 6.53178158737576650858416159924, 7.73092106136383601890635755188, 8.107355639526554076519300864650, 9.086973153737171919351246312674
