L(s) = 1 | − 2·4-s − 3·5-s + 7-s − 6·11-s − 13-s + 4·16-s − 4·19-s + 6·20-s − 3·23-s + 4·25-s − 2·28-s − 3·29-s + 5·31-s − 3·35-s − 7·37-s − 6·41-s − 10·43-s + 12·44-s − 6·47-s + 49-s + 2·52-s − 9·53-s + 18·55-s + 3·59-s − 7·61-s − 8·64-s + 3·65-s + ⋯ |
L(s) = 1 | − 4-s − 1.34·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 16-s − 0.917·19-s + 1.34·20-s − 0.625·23-s + 4/5·25-s − 0.377·28-s − 0.557·29-s + 0.898·31-s − 0.507·35-s − 1.15·37-s − 0.937·41-s − 1.52·43-s + 1.80·44-s − 0.875·47-s + 1/7·49-s + 0.277·52-s − 1.23·53-s + 2.42·55-s + 0.390·59-s − 0.896·61-s − 64-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−7.49553202033604929322937007665, −6.49957561357936923454065760600, −5.48861828138460437247099410441, −4.86782655190397896854010119964, −4.43418914689342265903546928509, −3.57745229619447584458496238979, −2.90609205456343971835068320772, −1.69645786272065183530046548431, 0, 0,
1.69645786272065183530046548431, 2.90609205456343971835068320772, 3.57745229619447584458496238979, 4.43418914689342265903546928509, 4.86782655190397896854010119964, 5.48861828138460437247099410441, 6.49957561357936923454065760600, 7.49553202033604929322937007665