| L(s) = 1 | − 2·3-s − 4·7-s + 9-s − 8·13-s − 8·19-s + 8·21-s + 4·27-s − 8·37-s + 16·39-s + 12·43-s − 2·49-s + 16·57-s − 20·61-s − 4·63-s − 28·67-s − 16·73-s + 32·79-s − 11·81-s + 32·91-s − 32·97-s − 28·103-s − 12·109-s + 16·111-s − 8·117-s − 6·121-s + 127-s − 24·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s − 2.21·13-s − 1.83·19-s + 1.74·21-s + 0.769·27-s − 1.31·37-s + 2.56·39-s + 1.82·43-s − 2/7·49-s + 2.11·57-s − 2.56·61-s − 0.503·63-s − 3.42·67-s − 1.87·73-s + 3.60·79-s − 1.22·81-s + 3.35·91-s − 3.24·97-s − 2.75·103-s − 1.14·109-s + 1.51·111-s − 0.739·117-s − 0.545·121-s + 0.0887·127-s − 2.11·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.125314960122643492265723439737, −7.72354144124853582700669789995, −7.14831674158858802892659992469, −6.62151311275959826478778900954, −6.55536659882703175830371730918, −5.77630218563910465834928061038, −5.62831154024710482280783911304, −4.65101662855925806243412279834, −4.64606420734121812424941498850, −3.86307675984348758158108701206, −2.85591890992545130759706287184, −2.75046174819041571949427472141, −1.69174310740343087998151716036, 0, 0,
1.69174310740343087998151716036, 2.75046174819041571949427472141, 2.85591890992545130759706287184, 3.86307675984348758158108701206, 4.64606420734121812424941498850, 4.65101662855925806243412279834, 5.62831154024710482280783911304, 5.77630218563910465834928061038, 6.55536659882703175830371730918, 6.62151311275959826478778900954, 7.14831674158858802892659992469, 7.72354144124853582700669789995, 8.125314960122643492265723439737
