| L(s) = 1 | − 2·3-s + 3·9-s − 8·11-s + 12·17-s − 10·25-s − 4·27-s + 16·33-s − 4·41-s + 16·43-s − 10·49-s − 24·51-s − 8·59-s + 8·67-s − 20·73-s + 20·75-s + 5·81-s + 24·83-s + 4·89-s − 12·97-s − 24·99-s − 24·107-s − 28·113-s + 26·121-s + 8·123-s + 127-s − 32·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 2.41·11-s + 2.91·17-s − 2·25-s − 0.769·27-s + 2.78·33-s − 0.624·41-s + 2.43·43-s − 1.42·49-s − 3.36·51-s − 1.04·59-s + 0.977·67-s − 2.34·73-s + 2.30·75-s + 5/9·81-s + 2.63·83-s + 0.423·89-s − 1.21·97-s − 2.41·99-s − 2.32·107-s − 2.63·113-s + 2.36·121-s + 0.721·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.355375973689939441990781827382, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −7.73185283482156320096864018539, −7.29744951786402483785793531451, −6.40139715881748782453956361829, −5.81271121311664380914911812156, −5.52928834758807112839354938677, −5.25073137830748218691985779847, −4.57667813854686016923499181308, −3.77490371552045331982722842815, −3.12662782741322079682173901720, −2.36547685273488160999702672858, −1.27109672072212401682622254955, 0,
1.27109672072212401682622254955, 2.36547685273488160999702672858, 3.12662782741322079682173901720, 3.77490371552045331982722842815, 4.57667813854686016923499181308, 5.25073137830748218691985779847, 5.52928834758807112839354938677, 5.81271121311664380914911812156, 6.40139715881748782453956361829, 7.29744951786402483785793531451, 7.73185283482156320096864018539, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 9.355375973689939441990781827382
