| L(s) = 1 | + 0.603·2-s + 3·3-s − 7.63·4-s + 14.1·5-s + 1.81·6-s + 7·7-s − 9.43·8-s + 9·9-s + 8.55·10-s − 54.0·11-s − 22.9·12-s − 81.7·13-s + 4.22·14-s + 42.5·15-s + 55.3·16-s − 21.9·17-s + 5.43·18-s − 124.·19-s − 108.·20-s + 21·21-s − 32.6·22-s + 23·23-s − 28.3·24-s + 76.1·25-s − 49.3·26-s + 27·27-s − 53.4·28-s + ⋯ |
| L(s) = 1 | + 0.213·2-s + 0.577·3-s − 0.954·4-s + 1.26·5-s + 0.123·6-s + 0.377·7-s − 0.416·8-s + 0.333·9-s + 0.270·10-s − 1.48·11-s − 0.551·12-s − 1.74·13-s + 0.0806·14-s + 0.732·15-s + 0.865·16-s − 0.313·17-s + 0.0711·18-s − 1.50·19-s − 1.21·20-s + 0.218·21-s − 0.316·22-s + 0.208·23-s − 0.240·24-s + 0.609·25-s − 0.372·26-s + 0.192·27-s − 0.360·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 0.603T + 8T^{2} \) |
| 5 | \( 1 - 14.1T + 125T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 81.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 21.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 69.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 405.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 69.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 587.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 380.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 244.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 649.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 629.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 682.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 691.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 753.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 866.T + 9.12e5T^{2} \) |
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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
−10.07798271158293590715198541324, −9.280112907713625439409248661431, −8.427391970526199535441037965547, −7.56867719332059679604714655064, −6.22726439634480730376827576196, −5.08330456345172683054042173728, −4.61332121931736578374510147761, −2.87335083598326043005280153929, −2.01094493051564107970424633082, 0,
2.01094493051564107970424633082, 2.87335083598326043005280153929, 4.61332121931736578374510147761, 5.08330456345172683054042173728, 6.22726439634480730376827576196, 7.56867719332059679604714655064, 8.427391970526199535441037965547, 9.280112907713625439409248661431, 10.07798271158293590715198541324
