L(s) = 1 | − 2·5-s + 4·7-s − 4·11-s + 2·13-s − 12·17-s − 8·19-s + 5·25-s − 2·29-s − 4·31-s − 8·35-s − 4·37-s − 2·41-s − 4·43-s − 8·47-s + 7·49-s + 20·53-s + 8·55-s + 4·59-s − 6·61-s − 4·65-s − 4·67-s − 32·71-s − 12·73-s − 16·77-s − 4·79-s − 12·83-s + 24·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.554·13-s − 2.91·17-s − 1.83·19-s + 25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s − 0.657·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 49-s + 2.74·53-s + 1.07·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 3.79·71-s − 1.40·73-s − 1.82·77-s − 0.450·79-s − 1.31·83-s + 2.60·85-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.731600469424218256517314059154, −8.488638825465717007412615424399, −7.968546495743276285892234566496, −7.55533611077282287995279534489, −7.13486764868327116963586724563, −7.05202789774082241386866214904, −6.21648940658876017122598339493, −6.18837101056719285953327028322, −5.52302718317988151036413742129, −4.86329120212920322314284534981, −4.72316216292546896078046326765, −4.48307130811217325809175245918, −3.93048013535419094775398629864, −3.57776422736899308372712736248, −2.74069323824921438756885998149, −2.33559924612471592659008956244, −1.92578464284909834771248956834, −1.37374890790367982153748551298, 0, 0,
1.37374890790367982153748551298, 1.92578464284909834771248956834, 2.33559924612471592659008956244, 2.74069323824921438756885998149, 3.57776422736899308372712736248, 3.93048013535419094775398629864, 4.48307130811217325809175245918, 4.72316216292546896078046326765, 4.86329120212920322314284534981, 5.52302718317988151036413742129, 6.18837101056719285953327028322, 6.21648940658876017122598339493, 7.05202789774082241386866214904, 7.13486764868327116963586724563, 7.55533611077282287995279534489, 7.968546495743276285892234566496, 8.488638825465717007412615424399, 8.731600469424218256517314059154