| L(s) = 1 | − 3-s + 2·7-s − 2·9-s − 3·11-s − 4·13-s + 3·17-s + 5·19-s − 2·21-s + 5·27-s − 2·31-s + 3·33-s − 2·37-s + 4·39-s − 3·41-s − 4·43-s − 12·47-s − 3·49-s − 3·51-s − 6·53-s − 5·57-s − 2·61-s − 4·63-s − 13·67-s − 12·71-s + 11·73-s − 6·77-s − 10·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.904·11-s − 1.10·13-s + 0.727·17-s + 1.14·19-s − 0.436·21-s + 0.962·27-s − 0.359·31-s + 0.522·33-s − 0.328·37-s + 0.640·39-s − 0.468·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.420·51-s − 0.824·53-s − 0.662·57-s − 0.256·61-s − 0.503·63-s − 1.58·67-s − 1.42·71-s + 1.28·73-s − 0.683·77-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−13.52942700993099, −12.87527705562080, −12.49658666138221, −11.89694791432739, −11.71386200410520, −11.16784420815525, −10.80194751428602, −10.15892727859441, −9.823728547444456, −9.406817243610037, −8.539387956362455, −8.331192549649400, −7.681753343161279, −7.407584311457936, −6.832752080773301, −6.144951571152375, −5.630706619020301, −5.109750871138102, −5.000560086348682, −4.387930737898722, −3.448920776196527, −2.983701574882919, −2.574271985728701, −1.643246633313582, −1.255262075652764, 0, 0,
1.255262075652764, 1.643246633313582, 2.574271985728701, 2.983701574882919, 3.448920776196527, 4.387930737898722, 5.000560086348682, 5.109750871138102, 5.630706619020301, 6.144951571152375, 6.832752080773301, 7.407584311457936, 7.681753343161279, 8.331192549649400, 8.539387956362455, 9.406817243610037, 9.823728547444456, 10.15892727859441, 10.80194751428602, 11.16784420815525, 11.71386200410520, 11.89694791432739, 12.49658666138221, 12.87527705562080, 13.52942700993099
