L(s) = 1 | − 7-s + 13-s + 2·19-s − 25-s + 2·31-s − 2·37-s + 2·43-s + 49-s + 61-s − 67-s − 2·73-s − 79-s − 91-s + 97-s − 103-s + 4·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 7-s + 13-s + 2·19-s − 25-s + 2·31-s − 2·37-s + 2·43-s + 49-s + 61-s − 67-s − 2·73-s − 79-s − 91-s + 97-s − 103-s + 4·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.034688834\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.034688834\) |
\(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.976460439726058876802478941422, −9.857554802837019139384160546371, −9.056943556033875290193898870540, −8.968348658327949297262650249439, −8.607493425990316238140279698133, −7.960091180218141000908313520896, −7.43021332229877915758758437671, −7.42429584257468520538659864693, −6.67852976714919991226496073411, −6.37565511981828672964346474963, −5.78394001834080959642183209820, −5.69412703741520531046856888022, −5.02757464026046468759624831698, −4.50009399754335994416246096190, −3.79245552020892662679042162058, −3.64716782261660270103531063276, −2.90003139280413342578176578766, −2.65918547005904147011194791027, −1.62883214494859894648940785575, −0.956525500722701893500926710313,
0.956525500722701893500926710313, 1.62883214494859894648940785575, 2.65918547005904147011194791027, 2.90003139280413342578176578766, 3.64716782261660270103531063276, 3.79245552020892662679042162058, 4.50009399754335994416246096190, 5.02757464026046468759624831698, 5.69412703741520531046856888022, 5.78394001834080959642183209820, 6.37565511981828672964346474963, 6.67852976714919991226496073411, 7.42429584257468520538659864693, 7.43021332229877915758758437671, 7.960091180218141000908313520896, 8.607493425990316238140279698133, 8.968348658327949297262650249439, 9.056943556033875290193898870540, 9.857554802837019139384160546371, 9.976460439726058876802478941422