L(s) = 1 | − 5-s − 7-s + 4·11-s + 2·13-s + 8·17-s + 6·19-s + 4·23-s + 25-s − 2·29-s − 4·31-s + 35-s − 8·37-s + 10·41-s + 10·43-s − 10·47-s + 49-s + 2·53-s − 4·55-s − 4·59-s + 4·61-s − 2·65-s + 10·67-s + 6·71-s + 6·73-s − 4·77-s + 8·79-s − 8·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.94·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.169·35-s − 1.31·37-s + 1.56·41-s + 1.52·43-s − 1.45·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s + 0.512·61-s − 0.248·65-s + 1.22·67-s + 0.712·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.474292180\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.474292180\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−16.61010813220752, −16.06611667223787, −15.64058773573859, −14.78655319163645, −14.29763835055251, −13.98494458179849, −13.10612114737931, −12.35411908538688, −12.19602493927304, −11.27104712328187, −11.03460839512113, −10.02437022941944, −9.501080277513116, −9.066036368923217, −8.239938898948910, −7.540569653205021, −7.097874485421190, −6.300081451300631, −5.578027461252717, −5.040728035866151, −3.794857469705622, −3.636699193712244, −2.801423053758577, −1.431834185284282, −0.8442548015094900,
0.8442548015094900, 1.431834185284282, 2.801423053758577, 3.636699193712244, 3.794857469705622, 5.040728035866151, 5.578027461252717, 6.300081451300631, 7.097874485421190, 7.540569653205021, 8.239938898948910, 9.066036368923217, 9.501080277513116, 10.02437022941944, 11.03460839512113, 11.27104712328187, 12.19602493927304, 12.35411908538688, 13.10612114737931, 13.98494458179849, 14.29763835055251, 14.78655319163645, 15.64058773573859, 16.06611667223787, 16.61010813220752