| L(s) = 1 | − 4·9-s − 2·11-s + 4·23-s − 8·25-s + 12·29-s − 20·37-s − 8·43-s − 16·53-s + 28·67-s − 12·71-s + 8·79-s + 7·81-s + 8·99-s + 16·107-s + 4·109-s − 4·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 0.603·11-s + 0.834·23-s − 8/5·25-s + 2.22·29-s − 3.28·37-s − 1.21·43-s − 2.19·53-s + 3.42·67-s − 1.42·71-s + 0.900·79-s + 7/9·81-s + 0.804·99-s + 1.54·107-s + 0.383·109-s − 0.376·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−7.51369766584857938216840087329, −7.35783842373057230890716087563, −6.83052361471556243095965304774, −6.59238542078930163139354131147, −6.15922458758044654938990820512, −6.02376962436180908926811308440, −5.30284755388887546564634349165, −5.28478285149040440348981654060, −4.77909006166690264420541907451, −4.76034706322327408007311720477, −3.89315987958130580367222571943, −3.60045712977443368128587111336, −3.18585893535117777496788999148, −3.03655523093383571432863747420, −2.28474418852789558295968223724, −2.20304934140257096980247925942, −1.51042290467858382595985635016, −0.973028907382995088418632044565, 0, 0,
0.973028907382995088418632044565, 1.51042290467858382595985635016, 2.20304934140257096980247925942, 2.28474418852789558295968223724, 3.03655523093383571432863747420, 3.18585893535117777496788999148, 3.60045712977443368128587111336, 3.89315987958130580367222571943, 4.76034706322327408007311720477, 4.77909006166690264420541907451, 5.28478285149040440348981654060, 5.30284755388887546564634349165, 6.02376962436180908926811308440, 6.15922458758044654938990820512, 6.59238542078930163139354131147, 6.83052361471556243095965304774, 7.35783842373057230890716087563, 7.51369766584857938216840087329
