| L(s) = 1 | − 2·5-s + 3·9-s + 2·11-s − 2·13-s − 6·17-s + 6·19-s − 8·23-s + 5·25-s + 4·29-s − 10·31-s + 6·37-s − 12·41-s + 8·43-s − 6·45-s + 2·47-s − 6·53-s − 4·55-s + 10·59-s + 2·61-s + 4·65-s − 10·67-s + 20·71-s − 2·73-s + 4·79-s − 12·83-s + 12·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 9-s + 0.603·11-s − 0.554·13-s − 1.45·17-s + 1.37·19-s − 1.66·23-s + 25-s + 0.742·29-s − 1.79·31-s + 0.986·37-s − 1.87·41-s + 1.21·43-s − 0.894·45-s + 0.291·47-s − 0.824·53-s − 0.539·55-s + 1.30·59-s + 0.256·61-s + 0.496·65-s − 1.22·67-s + 2.37·71-s − 0.234·73-s + 0.450·79-s − 1.31·83-s + 1.30·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633924242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633924242\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.040969491223379481320719380049, −8.685391360335856963348159839855, −8.414782190775123716792812197500, −7.71196150938442075257121876618, −7.63656342102172153565445716620, −7.25368151037490098794554295022, −6.72177742441423383515781299593, −6.70224677680748236299205269460, −5.94962541115913824875881112916, −5.68903555459744579793984496929, −4.96507792699715729739111225875, −4.69559890433362762840730282490, −4.34255421129253938789531488999, −3.86171020252907736804330476234, −3.52552853922392142885805388737, −3.07124704419659356778672779706, −2.17926442020241845369320320993, −2.01533046223716986893157225989, −1.17317768737656065340148594055, −0.46355845509049415574572055847,
0.46355845509049415574572055847, 1.17317768737656065340148594055, 2.01533046223716986893157225989, 2.17926442020241845369320320993, 3.07124704419659356778672779706, 3.52552853922392142885805388737, 3.86171020252907736804330476234, 4.34255421129253938789531488999, 4.69559890433362762840730282490, 4.96507792699715729739111225875, 5.68903555459744579793984496929, 5.94962541115913824875881112916, 6.70224677680748236299205269460, 6.72177742441423383515781299593, 7.25368151037490098794554295022, 7.63656342102172153565445716620, 7.71196150938442075257121876618, 8.414782190775123716792812197500, 8.685391360335856963348159839855, 9.040969491223379481320719380049
