| L(s) = 1 | − 2·7-s − 2·9-s − 8·11-s + 4·23-s + 4·29-s − 8·37-s + 12·43-s − 3·49-s + 8·53-s + 4·63-s − 28·67-s + 16·71-s + 16·77-s + 32·79-s − 5·81-s + 16·99-s + 20·107-s − 12·109-s − 32·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 2/3·9-s − 2.41·11-s + 0.834·23-s + 0.742·29-s − 1.31·37-s + 1.82·43-s − 3/7·49-s + 1.09·53-s + 0.503·63-s − 3.42·67-s + 1.89·71-s + 1.82·77-s + 3.60·79-s − 5/9·81-s + 1.60·99-s + 1.93·107-s − 1.14·109-s − 3.01·113-s + 2.36·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.72354144124853582700669789995, −7.20511912671126545376415987851, −6.62151311275959826478778900954, −6.36916259218775338867521337025, −5.62831154024710482280783911304, −5.51293166761659743969374174316, −5.00192751095038063419861633350, −4.64606420734121812424941498850, −3.89903784798328546362956718852, −3.34995357407111759567627632435, −2.75046174819041571949427472141, −2.68534013766286934612944284896, −1.93552250057563034818517220292, −0.788949000917484654700473342274, 0,
0.788949000917484654700473342274, 1.93552250057563034818517220292, 2.68534013766286934612944284896, 2.75046174819041571949427472141, 3.34995357407111759567627632435, 3.89903784798328546362956718852, 4.64606420734121812424941498850, 5.00192751095038063419861633350, 5.51293166761659743969374174316, 5.62831154024710482280783911304, 6.36916259218775338867521337025, 6.62151311275959826478778900954, 7.20511912671126545376415987851, 7.72354144124853582700669789995
