| L(s) = 1 | − 5-s − 7-s − 4·11-s + 6·13-s + 6·17-s + 4·19-s + 8·23-s + 25-s − 10·29-s + 4·31-s + 35-s − 6·37-s − 6·41-s + 4·43-s + 12·47-s + 49-s − 6·53-s + 4·55-s − 4·59-s − 2·61-s − 6·65-s + 4·67-s − 2·73-s + 4·77-s − 8·79-s + 12·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s + 1.66·13-s + 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.744·65-s + 0.488·67-s − 0.234·73-s + 0.455·77-s − 0.900·79-s + 1.31·83-s − 0.650·85-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
| \(L(1)\) |
\(\approx\) |
\(1.905733258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905733258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.65900969519856, −15.91631264506027, −15.54208707584706, −15.17078835168496, −14.21324267582104, −13.75491921019693, −13.06667627911194, −12.76068326987576, −11.94961682542354, −11.36946716861259, −10.71119451540046, −10.37024822656426, −9.461813026046350, −8.940410528211834, −8.230574807987841, −7.584316477751267, −7.173997421738530, −6.220237660857773, −5.493350118903788, −5.142940828639515, −3.987678959090347, −3.329181129501338, −2.896538982490151, −1.543839566749683, −0.6895183158063298,
0.6895183158063298, 1.543839566749683, 2.896538982490151, 3.329181129501338, 3.987678959090347, 5.142940828639515, 5.493350118903788, 6.220237660857773, 7.173997421738530, 7.584316477751267, 8.230574807987841, 8.940410528211834, 9.461813026046350, 10.37024822656426, 10.71119451540046, 11.36946716861259, 11.94961682542354, 12.76068326987576, 13.06667627911194, 13.75491921019693, 14.21324267582104, 15.17078835168496, 15.54208707584706, 15.91631264506027, 16.65900969519856
