L(s) = 1 | + 3-s − 7-s + 9-s + 6·11-s + 2·13-s + 4·17-s + 2·19-s − 21-s − 23-s + 27-s − 8·29-s + 6·31-s + 6·33-s − 2·37-s + 2·39-s − 10·41-s + 2·43-s + 2·47-s + 49-s + 4·51-s + 6·53-s + 2·57-s + 4·59-s + 6·61-s − 63-s + 2·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.970·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.48·29-s + 1.07·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s − 1.56·41-s + 0.304·43-s + 0.291·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.244·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.320896188\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.320896188\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.86554885816439, −13.46933736094274, −12.79578176200686, −12.31432008972246, −11.83002641644823, −11.47996488617635, −10.86956831096514, −10.14840538404555, −9.716485332526704, −9.417421409241185, −8.756289257834815, −8.438842411232679, −7.789846594587224, −7.211324106984737, −6.661678972750786, −6.338870200311300, −5.558577673569416, −5.147865142000304, −4.157026764718644, −3.777235325051336, −3.465406483593388, −2.709533389104673, −1.885107153615529, −1.315489702025182, −0.6801064411080611,
0.6801064411080611, 1.315489702025182, 1.885107153615529, 2.709533389104673, 3.465406483593388, 3.777235325051336, 4.157026764718644, 5.147865142000304, 5.558577673569416, 6.338870200311300, 6.661678972750786, 7.211324106984737, 7.789846594587224, 8.438842411232679, 8.756289257834815, 9.417421409241185, 9.716485332526704, 10.14840538404555, 10.86956831096514, 11.47996488617635, 11.83002641644823, 12.31432008972246, 12.79578176200686, 13.46933736094274, 13.86554885816439