| L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 17-s − 4·19-s − 8·23-s − 25-s + 27-s − 6·29-s + 4·33-s + 2·37-s − 2·39-s − 10·41-s − 4·43-s − 2·45-s − 51-s − 6·53-s − 8·55-s − 4·57-s + 4·59-s + 6·61-s + 4·65-s − 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.328·37-s − 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.140·51-s − 0.824·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.768·61-s + 0.496·65-s − 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6709003781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6709003781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.29273508695369, −12.78733525102909, −12.19165047442238, −11.95007370957775, −11.44423653188706, −11.04504768959321, −10.28936759802908, −9.910346884447297, −9.434843659548820, −8.870725311589705, −8.455793501019039, −7.943563528088051, −7.579517908967650, −7.003528340011052, −6.409169144199781, −6.132067334046194, −5.223416918066153, −4.687646281970889, −4.031359131220226, −3.779679362749343, −3.352501520977433, −2.357503971821026, −1.981763213826892, −1.315130389070702, −0.2246248920318965,
0.2246248920318965, 1.315130389070702, 1.981763213826892, 2.357503971821026, 3.352501520977433, 3.779679362749343, 4.031359131220226, 4.687646281970889, 5.223416918066153, 6.132067334046194, 6.409169144199781, 7.003528340011052, 7.579517908967650, 7.943563528088051, 8.455793501019039, 8.870725311589705, 9.434843659548820, 9.910346884447297, 10.28936759802908, 11.04504768959321, 11.44423653188706, 11.95007370957775, 12.19165047442238, 12.78733525102909, 13.29273508695369
