| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·7-s + 2·10-s − 4·13-s − 4·14-s − 4·16-s − 2·17-s − 2·20-s + 23-s − 4·25-s + 8·26-s + 4·28-s + 7·31-s + 8·32-s + 4·34-s − 2·35-s + 3·37-s − 8·41-s + 6·43-s − 2·46-s − 8·47-s − 3·49-s + 8·50-s − 8·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.755·7-s + 0.632·10-s − 1.10·13-s − 1.06·14-s − 16-s − 0.485·17-s − 0.447·20-s + 0.208·23-s − 4/5·25-s + 1.56·26-s + 0.755·28-s + 1.25·31-s + 1.41·32-s + 0.685·34-s − 0.338·35-s + 0.493·37-s − 1.24·41-s + 0.914·43-s − 0.294·46-s − 1.16·47-s − 3/7·49-s + 1.13·50-s − 1.10·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.461316816361475492652505870168, −8.571719581705043558005338128568, −7.927183238362238013907773101180, −7.36742097706937993051027630389, −6.42924768192751341652305216268, −5.01784321266906662288549345024, −4.27593882764012714082242634416, −2.65868050490226641918200240944, −1.51509492168516879981983515735, 0,
1.51509492168516879981983515735, 2.65868050490226641918200240944, 4.27593882764012714082242634416, 5.01784321266906662288549345024, 6.42924768192751341652305216268, 7.36742097706937993051027630389, 7.927183238362238013907773101180, 8.571719581705043558005338128568, 9.461316816361475492652505870168
