| L(s) = 1 | − 4·5-s + 6·7-s + 21·11-s − 22·13-s + 22·17-s + 42·23-s + 25·25-s − 34·29-s − 12·31-s − 24·35-s + 32·37-s + 13·41-s − 87·43-s + 6·47-s − 25·49-s + 104·53-s − 84·55-s + 93·59-s − 16·61-s + 88·65-s + 201·67-s − 50·73-s + 126·77-s − 48·79-s − 60·83-s − 88·85-s + 4·89-s + ⋯ |
| L(s) = 1 | − 4/5·5-s + 6/7·7-s + 1.90·11-s − 1.69·13-s + 1.29·17-s + 1.82·23-s + 25-s − 1.17·29-s − 0.387·31-s − 0.685·35-s + 0.864·37-s + 0.317·41-s − 2.02·43-s + 6/47·47-s − 0.510·49-s + 1.96·53-s − 1.52·55-s + 1.57·59-s − 0.262·61-s + 1.35·65-s + 3·67-s − 0.684·73-s + 1.63·77-s − 0.607·79-s − 0.722·83-s − 1.03·85-s + 4/89·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.464659200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.464659200\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T - 9 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 61 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T + 268 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 479 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T + 1117 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T + 315 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T + 1009 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 13 T - 1512 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 87 T + 4372 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 2221 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 93 T + 6364 T^{2} - 93 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 16 T - 3465 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 48 T + 7009 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 60 T + 8089 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 43 T - 7560 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−9.321225934684429716285618599832, −8.850164417711057183636050179466, −8.566859833018779856039181812779, −8.240508420474155094773693956237, −7.51316761685733782532480506111, −7.44684820950049599957018690340, −7.06715063984778903324115607743, −6.68438431915442554221123891706, −6.24248106514829062050533023182, −5.45531123601800199214074263938, −5.11912826698585960945338299994, −5.00799942554169740138416156974, −4.22088214996845874944135508257, −3.99504248375645229097175939058, −3.43161560636291110242762384270, −3.01271655993761625778422113806, −2.31339723604115157043443198650, −1.69013278800731521485486076880, −1.06153142644478737531182791887, −0.58687757752064177514790165416,
0.58687757752064177514790165416, 1.06153142644478737531182791887, 1.69013278800731521485486076880, 2.31339723604115157043443198650, 3.01271655993761625778422113806, 3.43161560636291110242762384270, 3.99504248375645229097175939058, 4.22088214996845874944135508257, 5.00799942554169740138416156974, 5.11912826698585960945338299994, 5.45531123601800199214074263938, 6.24248106514829062050533023182, 6.68438431915442554221123891706, 7.06715063984778903324115607743, 7.44684820950049599957018690340, 7.51316761685733782532480506111, 8.240508420474155094773693956237, 8.566859833018779856039181812779, 8.850164417711057183636050179466, 9.321225934684429716285618599832
