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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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
−10.52073741751256101393346395830, −9.609430582163951997198546913218, −8.723597096299050161071918865696, −7.59979113057100630916600030897, −6.57821510484149350613671583746, −5.91251718474522033883940002696, −4.71147624069866472572464249323, −3.63744901676668709329257902218, −2.12357132707472814840785367639, 0,
2.12357132707472814840785367639, 3.63744901676668709329257902218, 4.71147624069866472572464249323, 5.91251718474522033883940002696, 6.57821510484149350613671583746, 7.59979113057100630916600030897, 8.723597096299050161071918865696, 9.609430582163951997198546913218, 10.52073741751256101393346395830