| L(s) = 1 | − 2.23·2-s − 3-s + 2.98·4-s + 2.23·6-s + 7-s − 2.20·8-s + 9-s + 5.44·11-s − 2.98·12-s + 1.71·13-s − 2.23·14-s − 1.05·16-s − 17-s − 2.23·18-s − 0.444·19-s − 21-s − 12.1·22-s − 5.57·23-s + 2.20·24-s − 3.82·26-s − 27-s + 2.98·28-s + 1.30·29-s − 1.26·31-s + 6.75·32-s − 5.44·33-s + 2.23·34-s + ⋯ |
| L(s) = 1 | − 1.57·2-s − 0.577·3-s + 1.49·4-s + 0.911·6-s + 0.377·7-s − 0.779·8-s + 0.333·9-s + 1.64·11-s − 0.862·12-s + 0.474·13-s − 0.596·14-s − 0.262·16-s − 0.242·17-s − 0.526·18-s − 0.102·19-s − 0.218·21-s − 2.59·22-s − 1.16·23-s + 0.450·24-s − 0.749·26-s − 0.192·27-s + 0.564·28-s + 0.242·29-s − 0.227·31-s + 1.19·32-s − 0.947·33-s + 0.382·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 11 | \( 1 - 5.44T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 19 | \( 1 + 0.444T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 9.54T + 37T^{2} \) |
| 41 | \( 1 - 8.54T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−7.41937892934407181720432185437, −6.90235330603893283405116427824, −6.24144096668748190815268413534, −5.68752926405416584060035217318, −4.38763011634132434671420322050, −4.06347309706041797464556434317, −2.71869653832623867266752414121, −1.61305860688595658047155301805, −1.19642744857891475517848157584, 0,
1.19642744857891475517848157584, 1.61305860688595658047155301805, 2.71869653832623867266752414121, 4.06347309706041797464556434317, 4.38763011634132434671420322050, 5.68752926405416584060035217318, 6.24144096668748190815268413534, 6.90235330603893283405116427824, 7.41937892934407181720432185437
