| L(s) = 1 | + 4·5-s − 3·7-s − 4·11-s − 13-s + 8·17-s + 2·19-s − 4·23-s + 5·25-s − 4·31-s − 12·35-s − 18·37-s − 8·43-s + 12·47-s + 7·49-s + 16·53-s − 16·55-s − 4·59-s + 5·61-s − 4·65-s + 11·67-s + 16·71-s + 2·73-s + 12·77-s − 5·79-s − 8·83-s + 32·85-s − 24·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.13·7-s − 1.20·11-s − 0.277·13-s + 1.94·17-s + 0.458·19-s − 0.834·23-s + 25-s − 0.718·31-s − 2.02·35-s − 2.95·37-s − 1.21·43-s + 1.75·47-s + 49-s + 2.19·53-s − 2.15·55-s − 0.520·59-s + 0.640·61-s − 0.496·65-s + 1.34·67-s + 1.89·71-s + 0.234·73-s + 1.36·77-s − 0.562·79-s − 0.878·83-s + 3.47·85-s − 2.54·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.122366738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122366738\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 8 T - 19 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.02170884125391701651409685082, −9.480909892471938857558869721817, −9.333154307785100565864704958112, −8.638132438245300380460439515168, −8.332802935702372913146679334186, −7.83627492024691991556789911610, −7.29505925695317916880085097214, −6.86744122284269292709356725206, −6.71989860919864914066355616211, −5.80859836593850024484642033298, −5.66371053168327754620653049130, −5.36906978344460176773306938762, −5.24122196836954494326969907862, −4.18629914470806684932915321082, −3.62855663242470304394979906726, −3.21015025661757063488451733208, −2.67216340198268497461872986171, −2.09039873743732264497871344375, −1.64762828548782518679932198388, −0.58908120434667112569858788197,
0.58908120434667112569858788197, 1.64762828548782518679932198388, 2.09039873743732264497871344375, 2.67216340198268497461872986171, 3.21015025661757063488451733208, 3.62855663242470304394979906726, 4.18629914470806684932915321082, 5.24122196836954494326969907862, 5.36906978344460176773306938762, 5.66371053168327754620653049130, 5.80859836593850024484642033298, 6.71989860919864914066355616211, 6.86744122284269292709356725206, 7.29505925695317916880085097214, 7.83627492024691991556789911610, 8.332802935702372913146679334186, 8.638132438245300380460439515168, 9.333154307785100565864704958112, 9.480909892471938857558869721817, 10.02170884125391701651409685082
