L(s) = 1 | + 3-s + 9-s + 11-s + 7·13-s + 4·17-s − 19-s − 23-s + 27-s + 8·29-s + 6·31-s + 33-s − 3·37-s + 7·39-s + 9·41-s − 4·43-s + 3·47-s + 4·51-s − 53-s − 57-s − 12·59-s + 4·61-s + 12·67-s − 69-s − 14·71-s + 14·73-s + 4·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.94·13-s + 0.970·17-s − 0.229·19-s − 0.208·23-s + 0.192·27-s + 1.48·29-s + 1.07·31-s + 0.174·33-s − 0.493·37-s + 1.12·39-s + 1.40·41-s − 0.609·43-s + 0.437·47-s + 0.560·51-s − 0.137·53-s − 0.132·57-s − 1.56·59-s + 0.512·61-s + 1.46·67-s − 0.120·69-s − 1.66·71-s + 1.63·73-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.585610543\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.585610543\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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
−13.04528895417317, −12.32193281527134, −12.12830310341047, −11.55468803846567, −10.94510800925008, −10.57402897951070, −10.20646906206595, −9.477368097595938, −9.245725005681507, −8.560804723825618, −8.186959587185884, −7.997712620151231, −7.235335516656623, −6.644726580125326, −6.223048371781062, −5.890605342731188, −5.135381661813841, −4.577866189593333, −3.989952696211689, −3.570661177364130, −3.072060825278363, −2.499149051461589, −1.731329851011893, −1.121911713012287, −0.7204723680344991,
0.7204723680344991, 1.121911713012287, 1.731329851011893, 2.499149051461589, 3.072060825278363, 3.570661177364130, 3.989952696211689, 4.577866189593333, 5.135381661813841, 5.890605342731188, 6.223048371781062, 6.644726580125326, 7.235335516656623, 7.997712620151231, 8.186959587185884, 8.560804723825618, 9.245725005681507, 9.477368097595938, 10.20646906206595, 10.57402897951070, 10.94510800925008, 11.55468803846567, 12.12830310341047, 12.32193281527134, 13.04528895417317