| L(s) = 1 | + 2-s + 4-s + 8-s + 2·13-s + 16-s − 2·19-s + 2·26-s + 6·29-s − 8·31-s + 32-s + 4·37-s − 2·38-s − 6·41-s − 2·43-s − 6·47-s + 2·52-s − 6·53-s + 6·58-s − 12·59-s − 8·61-s − 8·62-s + 64-s − 2·67-s + 6·71-s + 2·73-s + 4·74-s − 2·76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.554·13-s + 1/4·16-s − 0.458·19-s + 0.392·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.657·37-s − 0.324·38-s − 0.937·41-s − 0.304·43-s − 0.875·47-s + 0.277·52-s − 0.824·53-s + 0.787·58-s − 1.56·59-s − 1.02·61-s − 1.01·62-s + 1/8·64-s − 0.244·67-s + 0.712·71-s + 0.234·73-s + 0.464·74-s − 0.229·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.78364863771278, −15.21559057595697, −14.63518376986205, −14.23949530208942, −13.55459792484691, −13.19286790888462, −12.55584647976552, −12.12461411349204, −11.46958791902760, −10.88928240000783, −10.54975772141290, −9.722919770224042, −9.218053994027875, −8.380301271488095, −8.039933588888420, −7.175769797412137, −6.674461633906498, −6.050814059001846, −5.520876626167442, −4.704980707076101, −4.288136235446067, −3.397922716547883, −2.964227723883698, −1.964414948690779, −1.313819608961961, 0,
1.313819608961961, 1.964414948690779, 2.964227723883698, 3.397922716547883, 4.288136235446067, 4.704980707076101, 5.520876626167442, 6.050814059001846, 6.674461633906498, 7.175769797412137, 8.039933588888420, 8.380301271488095, 9.218053994027875, 9.722919770224042, 10.54975772141290, 10.88928240000783, 11.46958791902760, 12.12461411349204, 12.55584647976552, 13.19286790888462, 13.55459792484691, 14.23949530208942, 14.63518376986205, 15.21559057595697, 15.78364863771278
