| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 11-s + 2·13-s − 15-s + 2·19-s + 2·21-s + 25-s + 27-s + 8·31-s + 33-s − 2·35-s + 2·37-s + 2·39-s + 2·43-s − 45-s − 3·49-s + 6·53-s − 55-s + 2·57-s − 12·59-s + 2·61-s + 2·63-s − 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.458·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.43·31-s + 0.174·33-s − 0.338·35-s + 0.328·37-s + 0.320·39-s + 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s − 0.134·55-s + 0.264·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-s − 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921580816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921580816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.55122966166715937397226474944, −9.582343210944653898479186916646, −8.645708418051842654222381354756, −8.033613975551371841147286921739, −7.19099618254400368447274970134, −6.09482854588561899430923472839, −4.83817962781524391786500846781, −3.96194649889300253224389286726, −2.81164589120574202804379175461, −1.33452028302080270852116357753,
1.33452028302080270852116357753, 2.81164589120574202804379175461, 3.96194649889300253224389286726, 4.83817962781524391786500846781, 6.09482854588561899430923472839, 7.19099618254400368447274970134, 8.033613975551371841147286921739, 8.645708418051842654222381354756, 9.582343210944653898479186916646, 10.55122966166715937397226474944
