| L(s) = 1 | + 4·5-s + 9-s + 8·11-s + 12·19-s + 5·25-s − 24·29-s + 4·31-s + 32·41-s + 4·45-s + 32·55-s + 8·59-s − 28·61-s + 16·89-s + 48·95-s + 8·99-s − 32·101-s − 28·109-s + 38·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 96·145-s + 149-s + 151-s + 16·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s + 2.41·11-s + 2.75·19-s + 25-s − 4.45·29-s + 0.718·31-s + 4.99·41-s + 0.596·45-s + 4.31·55-s + 1.04·59-s − 3.58·61-s + 1.69·89-s + 4.92·95-s + 0.804·99-s − 3.18·101-s − 2.68·109-s + 3.45·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.97·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.666453097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.666453097\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 78 T^{2} + 3875 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.17466968381565439307023756981, −5.90005658324623876628681226669, −5.83025544126487308994576808427, −5.70188717355671023038426728772, −5.43403797619505699826815701132, −5.40775090376249462953154215922, −5.05321881105947221911586007869, −4.72515909994994945672842145294, −4.59299321641505539406685428158, −4.19025614488811868146398182118, −4.13531291866282157797152923334, −3.97178550668080515087729535371, −3.56750328386962050365295764238, −3.56523953075540899365414786973, −3.31976402014669517966946932384, −2.88034595254221481010795903970, −2.72846656090350038755280325576, −2.35611428909548945271352882371, −2.21854360894033444636621109786, −1.95637795947263873429893611643, −1.45855051364898902012861049767, −1.39276657961476286445178085952, −1.20603940490055246637402810889, −0.999529305838686521582577084870, −0.34040283102022453510269112234,
0.34040283102022453510269112234, 0.999529305838686521582577084870, 1.20603940490055246637402810889, 1.39276657961476286445178085952, 1.45855051364898902012861049767, 1.95637795947263873429893611643, 2.21854360894033444636621109786, 2.35611428909548945271352882371, 2.72846656090350038755280325576, 2.88034595254221481010795903970, 3.31976402014669517966946932384, 3.56523953075540899365414786973, 3.56750328386962050365295764238, 3.97178550668080515087729535371, 4.13531291866282157797152923334, 4.19025614488811868146398182118, 4.59299321641505539406685428158, 4.72515909994994945672842145294, 5.05321881105947221911586007869, 5.40775090376249462953154215922, 5.43403797619505699826815701132, 5.70188717355671023038426728772, 5.83025544126487308994576808427, 5.90005658324623876628681226669, 6.17466968381565439307023756981
