| L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s − 3·7-s + 3·8-s − 2·9-s + 2·10-s + 3·11-s + 12-s − 6·13-s + 3·14-s + 2·15-s − 16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s + 3·21-s − 3·22-s − 4·23-s − 3·24-s − 25-s + 6·26-s + 5·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.654·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 83 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.82349410001976328687202362359, −12.30534590659785910801180963406, −11.77890434965261413314394451050, −10.17676635448238491848838037425, −9.457827669916618767543831706200, −8.118927993931557797577222055683, −6.97450299304430271706246705192, −5.31327024180315445602831381888, −3.65475901244453246589839654026, 0,
3.65475901244453246589839654026, 5.31327024180315445602831381888, 6.97450299304430271706246705192, 8.118927993931557797577222055683, 9.457827669916618767543831706200, 10.17676635448238491848838037425, 11.77890434965261413314394451050, 12.30534590659785910801180963406, 13.82349410001976328687202362359
