| L(s) = 1 | + 2·7-s + 14·19-s − 6·31-s + 8·37-s + 20·43-s − 11·49-s + 6·61-s − 20·67-s + 10·73-s − 24·79-s − 20·97-s − 4·103-s − 14·109-s − 18·121-s + 127-s + 131-s + 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 3.21·19-s − 1.07·31-s + 1.31·37-s + 3.04·43-s − 1.57·49-s + 0.768·61-s − 2.44·67-s + 1.17·73-s − 2.70·79-s − 2.03·97-s − 0.394·103-s − 1.34·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 2.42·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.803939831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.803939831\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 24 T^{2} + 287 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 36 T^{2} + 767 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 4 T^{2} - 2193 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 16 T^{2} - 2553 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 42 T^{2} - 1717 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 88 T^{2} - 177 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
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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
−7.43670066713073198658901004969, −7.34219447592412128936471237106, −6.96243514482671641660078505422, −6.79424738633053806223789997352, −6.74180495241961951572975876537, −6.07446430343482198848618268123, −5.88164196817968200826309463182, −5.78719686531878836303649301820, −5.52817429289512831921921649265, −5.50777035672276479235371154492, −5.04660747963882135909051330952, −4.71810395129548834197003327002, −4.60682683621401214865326235609, −4.35121478697173678711743405307, −3.98728877621667091530523756105, −3.77315166425405347126678405918, −3.44250531904788190475577079798, −2.93625720098955296498880892687, −2.85667905731729881285241079692, −2.81923699543565944240662170460, −2.12496372588200648486040085856, −1.71683730666216784679185480171, −1.46666538950011298566897024351, −1.01078842271763297194305842871, −0.58959884735531724501078841954,
0.58959884735531724501078841954, 1.01078842271763297194305842871, 1.46666538950011298566897024351, 1.71683730666216784679185480171, 2.12496372588200648486040085856, 2.81923699543565944240662170460, 2.85667905731729881285241079692, 2.93625720098955296498880892687, 3.44250531904788190475577079798, 3.77315166425405347126678405918, 3.98728877621667091530523756105, 4.35121478697173678711743405307, 4.60682683621401214865326235609, 4.71810395129548834197003327002, 5.04660747963882135909051330952, 5.50777035672276479235371154492, 5.52817429289512831921921649265, 5.78719686531878836303649301820, 5.88164196817968200826309463182, 6.07446430343482198848618268123, 6.74180495241961951572975876537, 6.79424738633053806223789997352, 6.96243514482671641660078505422, 7.34219447592412128936471237106, 7.43670066713073198658901004969
