L(s) = 1 | − 2·3-s + 3·9-s + 2·25-s − 4·27-s + 2·43-s + 49-s + 2·61-s − 4·75-s + 2·79-s + 5·81-s + 2·103-s + 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 4·183-s + ⋯ |
L(s) = 1 | − 2·3-s + 3·9-s + 2·25-s − 4·27-s + 2·43-s + 49-s + 2·61-s − 4·75-s + 2·79-s + 5·81-s + 2·103-s + 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 4·183-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6921214904\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6921214904\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−9.544499961811301252033295962602, −9.307738165366125611936190636818, −8.697860773880285245161734484930, −8.517056705031260870947939638418, −7.62629068224974818003695326494, −7.54095756129989694440267090462, −7.09315512680034443487363823646, −6.68656893715597963171413035078, −6.33250073310559861019456396622, −6.01671452798109505572198179147, −5.47336193062646480000072591607, −5.22973362863369123442195996369, −4.71369754546875008412714419970, −4.51129172026092433671530386940, −3.79655806724703038595366208855, −3.57152672678582470820627022690, −2.57142287944830164507923250819, −2.16652834924953483956165763342, −1.17699197708606636947380596305, −0.829449255343469210635015169323,
0.829449255343469210635015169323, 1.17699197708606636947380596305, 2.16652834924953483956165763342, 2.57142287944830164507923250819, 3.57152672678582470820627022690, 3.79655806724703038595366208855, 4.51129172026092433671530386940, 4.71369754546875008412714419970, 5.22973362863369123442195996369, 5.47336193062646480000072591607, 6.01671452798109505572198179147, 6.33250073310559861019456396622, 6.68656893715597963171413035078, 7.09315512680034443487363823646, 7.54095756129989694440267090462, 7.62629068224974818003695326494, 8.517056705031260870947939638418, 8.697860773880285245161734484930, 9.307738165366125611936190636818, 9.544499961811301252033295962602