| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s − 2·13-s + 6·19-s + 21-s − 27-s + 6·29-s + 10·31-s + 2·33-s + 2·39-s + 6·41-s − 8·43-s + 12·47-s + 49-s − 6·53-s − 6·57-s + 6·61-s − 63-s + 4·67-s + 6·71-s + 14·73-s + 2·77-s + 4·79-s + 81-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.37·19-s + 0.218·21-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.348·33-s + 0.320·39-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.794·57-s + 0.768·61-s − 0.125·63-s + 0.488·67-s + 0.712·71-s + 1.63·73-s + 0.227·77-s + 0.450·79-s + 1/9·81-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695873232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695873232\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.19437498043074, −14.35754833661496, −13.87108710226236, −13.49621943192515, −12.80793955149891, −12.19370831104671, −12.01349609970021, −11.29598665602356, −10.75770963702665, −10.09655458216116, −9.814658931742485, −9.246555675458156, −8.413773038252174, −7.889794803277604, −7.361610695527760, −6.643088633733183, −6.295012131841476, −5.392235370157513, −5.110118635551457, −4.417480865871328, −3.663279047399770, −2.835246010922212, −2.407765893072732, −1.199940897561578, −0.5707684224826152,
0.5707684224826152, 1.199940897561578, 2.407765893072732, 2.835246010922212, 3.663279047399770, 4.417480865871328, 5.110118635551457, 5.392235370157513, 6.295012131841476, 6.643088633733183, 7.361610695527760, 7.889794803277604, 8.413773038252174, 9.246555675458156, 9.814658931742485, 10.09655458216116, 10.75770963702665, 11.29598665602356, 12.01349609970021, 12.19370831104671, 12.80793955149891, 13.49621943192515, 13.87108710226236, 14.35754833661496, 15.19437498043074
