L(s) = 1 | + 32·2-s − 318·3-s + 1.02e3·4-s − 3.12e3·5-s − 1.01e4·6-s − 7.07e4·7-s + 3.27e4·8-s − 7.60e4·9-s − 1.00e5·10-s + 2.38e5·11-s − 3.25e5·12-s − 2.09e6·13-s − 2.26e6·14-s + 9.93e5·15-s + 1.04e6·16-s + 5.95e6·17-s − 2.43e6·18-s + 1.02e7·19-s − 3.20e6·20-s + 2.24e7·21-s + 7.62e6·22-s − 3.53e6·23-s − 1.04e7·24-s + 9.76e6·25-s − 6.71e7·26-s + 8.05e7·27-s − 7.24e7·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.755·3-s + 1/2·4-s − 0.447·5-s − 0.534·6-s − 1.59·7-s + 0.353·8-s − 0.429·9-s − 0.316·10-s + 0.446·11-s − 0.377·12-s − 1.56·13-s − 1.12·14-s + 0.337·15-s + 1/4·16-s + 1.01·17-s − 0.303·18-s + 0.946·19-s − 0.223·20-s + 1.20·21-s + 0.315·22-s − 0.114·23-s − 0.267·24-s + 1/5·25-s − 1.10·26-s + 1.07·27-s − 0.795·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
good | 3 | \( 1 + 106 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 10102 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 238272 T + p^{11} T^{2} \) |
| 13 | \( 1 + 2097478 T + p^{11} T^{2} \) |
| 17 | \( 1 - 5955546 T + p^{11} T^{2} \) |
| 19 | \( 1 - 10210820 T + p^{11} T^{2} \) |
| 23 | \( 1 + 3535758 T + p^{11} T^{2} \) |
| 29 | \( 1 + 139304850 T + p^{11} T^{2} \) |
| 31 | \( 1 + 101002348 T + p^{11} T^{2} \) |
| 37 | \( 1 + 524913814 T + p^{11} T^{2} \) |
| 41 | \( 1 - 284590422 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1253635078 T + p^{11} T^{2} \) |
| 47 | \( 1 + 216106434 T + p^{11} T^{2} \) |
| 53 | \( 1 + 4881275358 T + p^{11} T^{2} \) |
| 59 | \( 1 - 8692473300 T + p^{11} T^{2} \) |
| 61 | \( 1 - 3296491802 T + p^{11} T^{2} \) |
| 67 | \( 1 - 18275027966 T + p^{11} T^{2} \) |
| 71 | \( 1 + 13287447588 T + p^{11} T^{2} \) |
| 73 | \( 1 + 32505250798 T + p^{11} T^{2} \) |
| 79 | \( 1 - 9297455960 T + p^{11} T^{2} \) |
| 83 | \( 1 + 22741484838 T + p^{11} T^{2} \) |
| 89 | \( 1 + 93378882390 T + p^{11} T^{2} \) |
| 97 | \( 1 + 5811134014 T + p^{11} T^{2} \) |
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
−17.01618597135081896713273389127, −16.11601892396703700439948601753, −14.49506184906278364109210905204, −12.67429178840850196346152466421, −11.72100741358271812309889062364, −9.829838726570310491095976970921, −7.04509828040427304497400589835, −5.47434125415943491018676469462, −3.28219328272675841331927180697, 0,
3.28219328272675841331927180697, 5.47434125415943491018676469462, 7.04509828040427304497400589835, 9.829838726570310491095976970921, 11.72100741358271812309889062364, 12.67429178840850196346152466421, 14.49506184906278364109210905204, 16.11601892396703700439948601753, 17.01618597135081896713273389127