| L(s) = 1 | + 3-s + 2·7-s + 9-s + 11-s − 2·17-s + 8·19-s + 2·21-s − 2·23-s − 5·25-s + 27-s − 6·29-s + 33-s − 2·37-s + 2·41-s + 4·43-s − 6·47-s − 3·49-s − 2·51-s − 8·53-s + 8·57-s − 8·59-s − 4·61-s + 2·63-s + 12·67-s − 2·69-s − 10·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.417·23-s − 25-s + 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.875·47-s − 3/7·49-s − 0.280·51-s − 1.09·53-s + 1.05·57-s − 1.04·59-s − 0.512·61-s + 0.251·63-s + 1.46·67-s − 0.240·69-s − 1.18·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.615337035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.615337035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82668747256991712652889180897, −11.16632389782077890222538088411, −9.878059255196998497210886139393, −9.128822812513574728216194612598, −8.000137089983394792061903107301, −7.30353049322876396256497991972, −5.85067973388612859603664023961, −4.62272295792561209958505633181, −3.35636076808418408560877918112, −1.74917552990885925178691276263,
1.74917552990885925178691276263, 3.35636076808418408560877918112, 4.62272295792561209958505633181, 5.85067973388612859603664023961, 7.30353049322876396256497991972, 8.000137089983394792061903107301, 9.128822812513574728216194612598, 9.878059255196998497210886139393, 11.16632389782077890222538088411, 11.82668747256991712652889180897
