| 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
−8.964611739651684148490628994172, −8.258627059714335509637521091651, −7.35501093432362066723852819058, −6.42777574105788732928612276555, −5.37652489907251130162344018787, −4.88359239990992956705299543976, −3.50180763645388964696767481365, −2.22537825151505059931539706123, −1.31880122532995929599016682212, 0,
1.31880122532995929599016682212, 2.22537825151505059931539706123, 3.50180763645388964696767481365, 4.88359239990992956705299543976, 5.37652489907251130162344018787, 6.42777574105788732928612276555, 7.35501093432362066723852819058, 8.258627059714335509637521091651, 8.964611739651684148490628994172
