| L(s) = 1 | + 4·7-s − 2·19-s + 8·25-s + 20·29-s − 20·41-s + 24·43-s − 2·49-s − 20·53-s + 24·59-s + 16·61-s + 16·71-s + 12·73-s − 12·89-s + 16·107-s − 12·113-s + 4·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.458·19-s + 8/5·25-s + 3.71·29-s − 3.12·41-s + 3.65·43-s − 2/7·49-s − 2.74·53-s + 3.12·59-s + 2.04·61-s + 1.89·71-s + 1.40·73-s − 1.27·89-s + 1.54·107-s − 1.12·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.280209687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.280209687\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−9.794016899796135216771393209884, −9.467833675955558206485723992858, −8.681729725480174863270102956246, −8.529328387751128566742027973583, −8.245422034110757923736611122994, −8.042002093302562572123530272945, −7.33774017461619826058639057211, −6.85067603869651007030626842406, −6.56367515045245335973639234107, −6.28618836418052364133872891072, −5.36092677339483047373607846600, −5.18084179701541969211734125613, −4.61228716912658158000555029225, −4.58365400720302668090992328073, −3.76360301192382975194235527835, −3.23133839037633065696402891294, −2.49920238489264669785081430596, −2.23487811188569162255459932050, −1.24744177128096154219579840111, −0.877217553245966555519678683096,
0.877217553245966555519678683096, 1.24744177128096154219579840111, 2.23487811188569162255459932050, 2.49920238489264669785081430596, 3.23133839037633065696402891294, 3.76360301192382975194235527835, 4.58365400720302668090992328073, 4.61228716912658158000555029225, 5.18084179701541969211734125613, 5.36092677339483047373607846600, 6.28618836418052364133872891072, 6.56367515045245335973639234107, 6.85067603869651007030626842406, 7.33774017461619826058639057211, 8.042002093302562572123530272945, 8.245422034110757923736611122994, 8.529328387751128566742027973583, 8.681729725480174863270102956246, 9.467833675955558206485723992858, 9.794016899796135216771393209884
