| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 4·16-s − 2·25-s − 2·32-s − 4·50-s + 3·64-s + 4·67-s + 81-s − 2·100-s − 4·107-s + 2·121-s + 127-s + 6·128-s + 131-s + 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 167-s + 4·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 4·16-s − 2·25-s − 2·32-s − 4·50-s + 3·64-s + 4·67-s + 81-s − 2·100-s − 4·107-s + 2·121-s + 127-s + 6·128-s + 131-s + 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 167-s + 4·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8494899057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8494899057\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.338289662367561573731150199714, −8.217643887931389091092843722523, −8.068392786840979445537088349006, −7.75105913992276727584293806689, −7.18501078020764113703784744275, −7.15054050484099859451196718693, −6.85578042785168392983892937457, −6.50684546958193927497430349016, −6.20815541429223100426914943401, −6.16362379872484522322149410634, −5.79187376900272240215849062212, −5.44130708637096255446434180773, −5.29705043031103232911833173329, −5.19064242793329841487128014132, −4.81185762247012590249160734986, −4.31058511544820050170153889786, −4.24493844124242812060246730120, −4.07998359524572789148276891745, −3.48796433458912739743561166380, −3.46186598769586487547194133605, −3.27461445975603032562592412312, −2.54874371146741917521703212656, −2.28883595509610354560748598200, −2.13228944372364409470001681880, −1.16891556086109939487859211773,
1.16891556086109939487859211773, 2.13228944372364409470001681880, 2.28883595509610354560748598200, 2.54874371146741917521703212656, 3.27461445975603032562592412312, 3.46186598769586487547194133605, 3.48796433458912739743561166380, 4.07998359524572789148276891745, 4.24493844124242812060246730120, 4.31058511544820050170153889786, 4.81185762247012590249160734986, 5.19064242793329841487128014132, 5.29705043031103232911833173329, 5.44130708637096255446434180773, 5.79187376900272240215849062212, 6.16362379872484522322149410634, 6.20815541429223100426914943401, 6.50684546958193927497430349016, 6.85578042785168392983892937457, 7.15054050484099859451196718693, 7.18501078020764113703784744275, 7.75105913992276727584293806689, 8.068392786840979445537088349006, 8.217643887931389091092843722523, 8.338289662367561573731150199714
