| L(s) = 1 | + 2·3-s + 9-s + 4·11-s − 2·13-s − 2·19-s − 4·23-s − 4·27-s − 10·29-s + 4·31-s + 8·33-s − 2·37-s − 4·39-s + 12·41-s + 4·43-s − 4·47-s + 2·53-s − 4·57-s − 10·59-s + 6·61-s − 4·67-s − 8·69-s + 12·71-s − 4·73-s + 4·79-s − 11·81-s + 14·83-s − 20·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.458·19-s − 0.834·23-s − 0.769·27-s − 1.85·29-s + 0.718·31-s + 1.39·33-s − 0.328·37-s − 0.640·39-s + 1.87·41-s + 0.609·43-s − 0.583·47-s + 0.274·53-s − 0.529·57-s − 1.30·59-s + 0.768·61-s − 0.488·67-s − 0.963·69-s + 1.42·71-s − 0.468·73-s + 0.450·79-s − 1.22·81-s + 1.53·83-s − 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.37451321625524, −13.81583277115388, −13.45205973855810, −12.81590196387313, −12.28447931374471, −11.92534511750987, −11.12125646120210, −10.95697359271119, −10.02432740635191, −9.538069727132350, −9.264773827461185, −8.792388771539033, −8.120212064030487, −7.781791476711423, −7.212707072055390, −6.628660973403242, −5.968996773556725, −5.558660117104231, −4.595644207101141, −4.120542370775150, −3.643548290829869, −3.055276873126778, −2.242067463130055, −1.959392808883146, −1.043196163519239, 0,
1.043196163519239, 1.959392808883146, 2.242067463130055, 3.055276873126778, 3.643548290829869, 4.120542370775150, 4.595644207101141, 5.558660117104231, 5.968996773556725, 6.628660973403242, 7.212707072055390, 7.781791476711423, 8.120212064030487, 8.792388771539033, 9.264773827461185, 9.538069727132350, 10.02432740635191, 10.95697359271119, 11.12125646120210, 11.92534511750987, 12.28447931374471, 12.81590196387313, 13.45205973855810, 13.81583277115388, 14.37451321625524
