| L(s) = 1 | + 2.05·2-s − 3.75·4-s + 14.5·5-s − 24.2·8-s + 29.8·10-s + 29.8·11-s − 13.3·13-s − 19.7·16-s − 64.2·17-s − 110.·19-s − 54.5·20-s + 61.3·22-s + 19.3·23-s + 85.3·25-s − 27.4·26-s − 111.·29-s + 192.·31-s + 152.·32-s − 132.·34-s − 71.5·37-s − 227.·38-s − 351.·40-s − 277.·41-s − 178.·43-s − 112.·44-s + 39.8·46-s − 531.·47-s + ⋯ |
| L(s) = 1 | + 0.728·2-s − 0.469·4-s + 1.29·5-s − 1.07·8-s + 0.944·10-s + 0.817·11-s − 0.284·13-s − 0.309·16-s − 0.917·17-s − 1.33·19-s − 0.609·20-s + 0.594·22-s + 0.175·23-s + 0.682·25-s − 0.207·26-s − 0.711·29-s + 1.11·31-s + 0.844·32-s − 0.667·34-s − 0.317·37-s − 0.970·38-s − 1.38·40-s − 1.05·41-s − 0.633·43-s − 0.383·44-s + 0.127·46-s − 1.65·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.05T + 8T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 11 | \( 1 - 29.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 64.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 277.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 178.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 722.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 663.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 608.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 976.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 791.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 935.T + 9.12e5T^{2} \) |
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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
−8.921312090293477923452914581645, −8.315671052795813534850172720028, −6.68513392244081608869745440802, −6.37309392765520623292389801405, −5.39366283885437401581943205165, −4.65583043705429639970561919243, −3.78121247416154817245925447508, −2.59047028250162005532818507251, −1.60981621940311158152830876499, 0,
1.60981621940311158152830876499, 2.59047028250162005532818507251, 3.78121247416154817245925447508, 4.65583043705429639970561919243, 5.39366283885437401581943205165, 6.37309392765520623292389801405, 6.68513392244081608869745440802, 8.315671052795813534850172720028, 8.921312090293477923452914581645
