| L(s) = 1 | − 4·3-s + 4·5-s + 6·7-s + 6·9-s − 2·11-s − 16·15-s + 2·17-s − 6·19-s − 24·21-s + 2·23-s + 11·25-s + 4·27-s + 14·29-s + 8·33-s + 24·35-s + 24·45-s + 14·47-s + 18·49-s − 8·51-s + 16·53-s − 8·55-s + 24·57-s + 6·59-s + 2·61-s + 36·63-s − 8·69-s + 6·73-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 2.26·7-s + 2·9-s − 0.603·11-s − 4.13·15-s + 0.485·17-s − 1.37·19-s − 5.23·21-s + 0.417·23-s + 11/5·25-s + 0.769·27-s + 2.59·29-s + 1.39·33-s + 4.05·35-s + 3.57·45-s + 2.04·47-s + 18/7·49-s − 1.12·51-s + 2.19·53-s − 1.07·55-s + 3.17·57-s + 0.781·59-s + 0.256·61-s + 4.53·63-s − 0.963·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.575788268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575788268\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−10.71939659000990607735654071823, −10.58492815345916074848767081043, −10.11564257766476204471775399216, −9.956232091402147762700204742524, −8.789003204104276890135357857383, −8.675000093834449598913181619331, −8.384707588473095304354851238452, −7.64372194904857167935923136168, −6.84697119957715395753941088827, −6.78124025164253148360976191081, −5.96366063611134929173051975689, −5.81136509128689591171595016118, −5.35469621317564831575329942606, −5.05562347420124196102882738936, −4.64405349424528611062344707573, −4.21409684340139055725408044568, −2.60426345556077271494856877740, −2.42843520072922074682998445031, −1.35578562905217315664829285088, −0.915607802733132282268634140246,
0.915607802733132282268634140246, 1.35578562905217315664829285088, 2.42843520072922074682998445031, 2.60426345556077271494856877740, 4.21409684340139055725408044568, 4.64405349424528611062344707573, 5.05562347420124196102882738936, 5.35469621317564831575329942606, 5.81136509128689591171595016118, 5.96366063611134929173051975689, 6.78124025164253148360976191081, 6.84697119957715395753941088827, 7.64372194904857167935923136168, 8.384707588473095304354851238452, 8.675000093834449598913181619331, 8.789003204104276890135357857383, 9.956232091402147762700204742524, 10.11564257766476204471775399216, 10.58492815345916074848767081043, 10.71939659000990607735654071823
