| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 2·13-s − 2·15-s − 17-s − 6·19-s + 21-s − 8·23-s − 25-s − 27-s − 4·29-s − 2·35-s + 10·37-s + 2·39-s + 10·41-s + 8·43-s + 2·45-s + 8·47-s + 49-s + 51-s + 6·53-s + 6·57-s + 4·59-s − 8·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 1.37·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.338·35-s + 1.64·37-s + 0.320·39-s + 1.56·41-s + 1.21·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 0.794·57-s + 0.520·59-s − 1.02·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.390802328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390802328\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.967534058776462620796826987336, −7.42593018296492102848997665665, −6.39144411162584512587614181804, −6.05511858868767545804607064640, −5.48087740960946979158554221775, −4.38900191320476067910239169923, −3.93341427342984347367353728718, −2.45311772687458267392866671595, −2.06192578108688719125291261421, −0.62103412295080556702762325135,
0.62103412295080556702762325135, 2.06192578108688719125291261421, 2.45311772687458267392866671595, 3.93341427342984347367353728718, 4.38900191320476067910239169923, 5.48087740960946979158554221775, 6.05511858868767545804607064640, 6.39144411162584512587614181804, 7.42593018296492102848997665665, 7.967534058776462620796826987336
