L(s) = 1 | − 3-s + 2·5-s + 6·11-s + 6·13-s − 2·15-s + 4·17-s − 5·19-s + 4·23-s + 5·25-s + 27-s − 8·29-s + 7·31-s − 6·33-s + 9·37-s − 6·39-s + 4·41-s − 2·43-s + 2·47-s − 4·51-s − 8·53-s + 12·55-s + 5·57-s + 10·61-s + 12·65-s + 15·67-s − 4·69-s − 12·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.80·11-s + 1.66·13-s − 0.516·15-s + 0.970·17-s − 1.14·19-s + 0.834·23-s + 25-s + 0.192·27-s − 1.48·29-s + 1.25·31-s − 1.04·33-s + 1.47·37-s − 0.960·39-s + 0.624·41-s − 0.304·43-s + 0.291·47-s − 0.560·51-s − 1.09·53-s + 1.61·55-s + 0.662·57-s + 1.28·61-s + 1.48·65-s + 1.83·67-s − 0.481·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.023415167\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.023415167\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.762462760676157771304165840511, −9.731521763780025405322187399186, −9.094161503018299176577809475718, −8.897411213154396470175264085479, −8.324999790363514464437235168560, −8.203342001607343311746621593652, −7.18254928706451394834409851677, −7.14080934452002087315733315404, −6.39287676159782877920661980822, −6.21264154803786597600178023908, −5.84156156641845904224464611681, −5.61996560163064465773198465360, −4.61854071373634910023436186944, −4.57511630842210939148503411753, −3.73810090809893709000472987622, −3.50634807801491923677891080400, −2.74893910701963816799709895925, −1.99551513966424161448372955339, −1.26791861342647292081381636866, −0.951383593692832491129476640460,
0.951383593692832491129476640460, 1.26791861342647292081381636866, 1.99551513966424161448372955339, 2.74893910701963816799709895925, 3.50634807801491923677891080400, 3.73810090809893709000472987622, 4.57511630842210939148503411753, 4.61854071373634910023436186944, 5.61996560163064465773198465360, 5.84156156641845904224464611681, 6.21264154803786597600178023908, 6.39287676159782877920661980822, 7.14080934452002087315733315404, 7.18254928706451394834409851677, 8.203342001607343311746621593652, 8.324999790363514464437235168560, 8.897411213154396470175264085479, 9.094161503018299176577809475718, 9.731521763780025405322187399186, 9.762462760676157771304165840511