| L(s) = 1 | − 2.37·2-s + 3·3-s − 2.35·4-s − 18.5·5-s − 7.12·6-s + 7·7-s + 24.6·8-s + 9·9-s + 44.1·10-s + 29.5·11-s − 7.05·12-s − 64.1·13-s − 16.6·14-s − 55.7·15-s − 39.6·16-s + 57.2·17-s − 21.3·18-s + 93.4·19-s + 43.7·20-s + 21·21-s − 70.3·22-s + 23·23-s + 73.8·24-s + 220.·25-s + 152.·26-s + 27·27-s − 16.4·28-s + ⋯ |
| L(s) = 1 | − 0.840·2-s + 0.577·3-s − 0.293·4-s − 1.66·5-s − 0.485·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s + 1.39·10-s + 0.811·11-s − 0.169·12-s − 1.36·13-s − 0.317·14-s − 0.959·15-s − 0.619·16-s + 0.817·17-s − 0.280·18-s + 1.12·19-s + 0.488·20-s + 0.218·21-s − 0.681·22-s + 0.208·23-s + 0.627·24-s + 1.76·25-s + 1.14·26-s + 0.192·27-s − 0.111·28-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 | \( 1 + 2.37T + 8T^{2} \) |
| 5 | \( 1 + 18.5T + 125T^{2} \) |
| 11 | \( 1 - 29.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 76.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 68.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 561.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 481.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 105.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 870.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 526.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 79.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 321.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45T + 7.04e5T^{2} \) |
| 97 | \( 1 + 97.0T + 9.12e5T^{2} \) |
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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
−9.758423380492150901995286913219, −9.280892300021405295321077062832, −8.089379337365609263246206477617, −7.78154051063068777752420988036, −7.03915325318762522792946858649, −5.02670556465925094477465878869, −4.19698455101644153584126207306, −3.18912361234917546322511371070, −1.31948511194252512158396774561, 0,
1.31948511194252512158396774561, 3.18912361234917546322511371070, 4.19698455101644153584126207306, 5.02670556465925094477465878869, 7.03915325318762522792946858649, 7.78154051063068777752420988036, 8.089379337365609263246206477617, 9.280892300021405295321077062832, 9.758423380492150901995286913219
