| L(s) = 1 | + 4·11-s − 4·43-s − 4·53-s − 4·67-s + 81-s + 4·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·11-s − 4·43-s − 4·53-s − 4·67-s + 81-s + 4·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.618494379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618494379\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.78455456199932045878558263786, −6.43128032781403388384822115981, −6.34978255089868026487900151360, −6.17295131574672602007520387326, −6.01391923961920037128300571150, −5.84436109040130761113841306201, −5.59075646136150993324910360243, −5.11706896288769908397277154076, −4.88833739359896774435953959533, −4.69109466994569373153976826800, −4.67065651444874476297985607706, −4.37322042557541065139711921262, −4.20756965434041532849194256647, −3.78288295370457570616589227006, −3.63696915767250977084305921486, −3.37410373130829752072206870480, −3.30123222172558483690853473092, −3.06473609289237927131530240221, −2.80856031165082129564887514783, −2.20330101542419209099883202251, −1.74746769945061205058032668569, −1.71049858305065369401342424399, −1.55415969559917892806676621842, −1.30331451197242421052466012012, −0.67000012351427312704590870816,
0.67000012351427312704590870816, 1.30331451197242421052466012012, 1.55415969559917892806676621842, 1.71049858305065369401342424399, 1.74746769945061205058032668569, 2.20330101542419209099883202251, 2.80856031165082129564887514783, 3.06473609289237927131530240221, 3.30123222172558483690853473092, 3.37410373130829752072206870480, 3.63696915767250977084305921486, 3.78288295370457570616589227006, 4.20756965434041532849194256647, 4.37322042557541065139711921262, 4.67065651444874476297985607706, 4.69109466994569373153976826800, 4.88833739359896774435953959533, 5.11706896288769908397277154076, 5.59075646136150993324910360243, 5.84436109040130761113841306201, 6.01391923961920037128300571150, 6.17295131574672602007520387326, 6.34978255089868026487900151360, 6.43128032781403388384822115981, 6.78455456199932045878558263786
