| L(s) = 1 | + 3·3-s + 2·5-s + 6·9-s − 4·11-s + 3·13-s + 6·15-s + 7·17-s + 5·19-s − 4·23-s + 5·25-s + 9·27-s + 29-s + 6·31-s − 12·33-s − 11·37-s + 9·39-s − 9·41-s − 5·43-s + 12·45-s − 6·47-s + 21·51-s − 3·53-s − 8·55-s + 15·57-s + 14·59-s − 6·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 2·9-s − 1.20·11-s + 0.832·13-s + 1.54·15-s + 1.69·17-s + 1.14·19-s − 0.834·23-s + 25-s + 1.73·27-s + 0.185·29-s + 1.07·31-s − 2.08·33-s − 1.80·37-s + 1.44·39-s − 1.40·41-s − 0.762·43-s + 1.78·45-s − 0.875·47-s + 2.94·51-s − 0.412·53-s − 1.07·55-s + 1.98·57-s + 1.82·59-s − 0.768·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.444662300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.444662300\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.816832904352433644236642448105, −9.120807777214719620873676188780, −8.496231494552671455275119486478, −8.301774004122221458064652006628, −8.166892114151927444655121758420, −7.69007739914466243808242090016, −7.16951288624935488429464463372, −6.84483298123254027906115473969, −6.42613742218254134846892595742, −5.75089422604068289041166831504, −5.31365696501857716759121564615, −5.18742021201644664951498607798, −4.52135394542479127503723948004, −3.82869510926641463380100602001, −3.37462675618557716701940182390, −3.09454632223976632405438671401, −2.73960157951384522743806997312, −1.89777830414191377740275437163, −1.68309261561478016638137928214, −0.882924356690714378506897726111,
0.882924356690714378506897726111, 1.68309261561478016638137928214, 1.89777830414191377740275437163, 2.73960157951384522743806997312, 3.09454632223976632405438671401, 3.37462675618557716701940182390, 3.82869510926641463380100602001, 4.52135394542479127503723948004, 5.18742021201644664951498607798, 5.31365696501857716759121564615, 5.75089422604068289041166831504, 6.42613742218254134846892595742, 6.84483298123254027906115473969, 7.16951288624935488429464463372, 7.69007739914466243808242090016, 8.166892114151927444655121758420, 8.301774004122221458064652006628, 8.496231494552671455275119486478, 9.120807777214719620873676188780, 9.816832904352433644236642448105
