| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s − 13-s + 14-s − 16-s + 2·17-s − 4·19-s + 6·23-s − 5·25-s + 26-s + 28-s − 4·29-s − 5·32-s − 2·34-s − 10·37-s + 4·38-s + 12·41-s − 4·43-s − 6·46-s − 10·47-s + 49-s + 5·50-s + 52-s − 12·53-s − 3·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 0.277·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.196·26-s + 0.188·28-s − 0.742·29-s − 0.883·32-s − 0.342·34-s − 1.64·37-s + 0.648·38-s + 1.87·41-s − 0.609·43-s − 0.884·46-s − 1.45·47-s + 1/7·49-s + 0.707·50-s + 0.138·52-s − 1.64·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.587082283020044926918864997109, −9.159974617629976035930287366354, −8.164818245246782084230845868645, −7.47897633905458451916564903277, −6.45118986308248039122074236765, −5.31391085932205176006883258874, −4.37337084320337606308675064790, −3.24820983484805712022153239284, −1.65801382237619163464460077610, 0,
1.65801382237619163464460077610, 3.24820983484805712022153239284, 4.37337084320337606308675064790, 5.31391085932205176006883258874, 6.45118986308248039122074236765, 7.47897633905458451916564903277, 8.164818245246782084230845868645, 9.159974617629976035930287366354, 9.587082283020044926918864997109
