| L(s) = 1 | − 2-s + 4-s − 3.73·5-s − 8-s + 3.73·10-s − 4.19·11-s − 0.464·13-s + 16-s − 7·17-s + 2.73·19-s − 3.73·20-s + 4.19·22-s + 6.19·23-s + 8.92·25-s + 0.464·26-s + 8.46·29-s + 2.19·31-s − 32-s + 7·34-s − 6.66·37-s − 2.73·38-s + 3.73·40-s − 9.46·41-s + 5.46·43-s − 4.19·44-s − 6.19·46-s − 1.26·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.66·5-s − 0.353·8-s + 1.18·10-s − 1.26·11-s − 0.128·13-s + 0.250·16-s − 1.69·17-s + 0.626·19-s − 0.834·20-s + 0.894·22-s + 1.29·23-s + 1.78·25-s + 0.0910·26-s + 1.57·29-s + 0.394·31-s − 0.176·32-s + 1.20·34-s − 1.09·37-s − 0.443·38-s + 0.590·40-s − 1.47·41-s + 0.833·43-s − 0.632·44-s − 0.913·46-s − 0.184·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−7.58677431120726144502780469001, −6.96085324499862502445758868732, −6.52901084404543664372241442185, −5.11317588919856675867562227440, −4.81635831826095306745883721760, −3.79465714330597531456444646397, −3.02713427318853633827314235432, −2.32341493272433991614020085396, −0.869569907460402242523599031835, 0,
0.869569907460402242523599031835, 2.32341493272433991614020085396, 3.02713427318853633827314235432, 3.79465714330597531456444646397, 4.81635831826095306745883721760, 5.11317588919856675867562227440, 6.52901084404543664372241442185, 6.96085324499862502445758868732, 7.58677431120726144502780469001
