| L(s) = 1 | − 8·5-s − 34·13-s + 14·17-s + 39·25-s − 94·37-s + 160·41-s + 34·53-s + 240·61-s + 272·65-s − 206·73-s − 81·81-s − 112·85-s − 14·97-s + 396·101-s − 194·113-s − 242·121-s − 112·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 578·169-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 2.61·13-s + 0.823·17-s + 1.55·25-s − 2.54·37-s + 3.90·41-s + 0.641·53-s + 3.93·61-s + 4.18·65-s − 2.82·73-s − 81-s − 1.31·85-s − 0.144·97-s + 3.92·101-s − 1.71·113-s − 2·121-s − 0.895·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.42·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8232023196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8232023196\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 8 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )( 1 + 78 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.82671548808803046709025181606, −12.27793305373324612995885856158, −11.90549422290975687265989816487, −11.63807752129268011768333407435, −11.00748677334921265502286097673, −10.21296172549479647578563963136, −10.10708646442063944472817739810, −9.288574210560027430377715916440, −8.799190185937891743962117613846, −8.099069955660332990069149443385, −7.57980161974997963906567376394, −7.24230357182732721034206893491, −6.90384934763302524516276848973, −5.69816290538786908188894351379, −5.19750001638651342043084405508, −4.46503362035632265470202646384, −3.96665623126252917653167449733, −3.07014799261480417326737520363, −2.31946572742306242166285091625, −0.56241810294793935787052276502,
0.56241810294793935787052276502, 2.31946572742306242166285091625, 3.07014799261480417326737520363, 3.96665623126252917653167449733, 4.46503362035632265470202646384, 5.19750001638651342043084405508, 5.69816290538786908188894351379, 6.90384934763302524516276848973, 7.24230357182732721034206893491, 7.57980161974997963906567376394, 8.099069955660332990069149443385, 8.799190185937891743962117613846, 9.288574210560027430377715916440, 10.10708646442063944472817739810, 10.21296172549479647578563963136, 11.00748677334921265502286097673, 11.63807752129268011768333407435, 11.90549422290975687265989816487, 12.27793305373324612995885856158, 12.82671548808803046709025181606
