| L(s) = 1 | − 2·5-s − 3·7-s − 6·11-s + 3·13-s − 4·17-s − 6·19-s − 6·23-s + 5·25-s + 8·29-s + 6·35-s + 14·37-s − 8·41-s + 12·43-s − 6·47-s + 7·49-s + 8·53-s + 12·55-s − 6·59-s + 61-s − 6·65-s + 3·67-s + 24·71-s − 30·73-s + 18·77-s − 9·79-s + 12·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s − 1.80·11-s + 0.832·13-s − 0.970·17-s − 1.37·19-s − 1.25·23-s + 25-s + 1.48·29-s + 1.01·35-s + 2.30·37-s − 1.24·41-s + 1.82·43-s − 0.875·47-s + 49-s + 1.09·53-s + 1.61·55-s − 0.781·59-s + 0.128·61-s − 0.744·65-s + 0.366·67-s + 2.84·71-s − 3.51·73-s + 2.05·77-s − 1.01·79-s + 1.31·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3845067409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3845067409\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.011996121292348459824753466404, −8.601966968034222253608195119857, −8.131653467419613408023727006837, −8.126885370068648381292660471126, −7.69139172072118987836415953332, −7.11023939657461467003850127107, −6.74721077846424217419143685184, −6.36515226335793498346682807937, −6.16043664778174909414480651459, −5.54893302847208674734349387869, −5.28163119026811446528389108271, −4.46231438940450273872916865404, −4.24449646441145338824423210098, −4.10052589065827996560663852422, −3.30885124232416514397037093332, −2.82073206755928660023093256695, −2.59741191012505816484055479132, −2.04978175212343463560244884481, −1.01838773833009563796825520177, −0.23429718909680251568145510846,
0.23429718909680251568145510846, 1.01838773833009563796825520177, 2.04978175212343463560244884481, 2.59741191012505816484055479132, 2.82073206755928660023093256695, 3.30885124232416514397037093332, 4.10052589065827996560663852422, 4.24449646441145338824423210098, 4.46231438940450273872916865404, 5.28163119026811446528389108271, 5.54893302847208674734349387869, 6.16043664778174909414480651459, 6.36515226335793498346682807937, 6.74721077846424217419143685184, 7.11023939657461467003850127107, 7.69139172072118987836415953332, 8.126885370068648381292660471126, 8.131653467419613408023727006837, 8.601966968034222253608195119857, 9.011996121292348459824753466404
