| L(s) = 1 | + 3·3-s − 5-s − 3·7-s + 6·9-s + 4·11-s + 4·13-s − 3·15-s + 4·17-s + 4·19-s − 9·21-s + 7·23-s + 9·27-s + 9·29-s − 6·31-s + 12·33-s + 3·35-s + 4·37-s + 12·39-s − 9·41-s − 4·43-s − 6·45-s + 3·47-s + 7·49-s + 12·51-s + 12·53-s − 4·55-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s + 1.20·11-s + 1.10·13-s − 0.774·15-s + 0.970·17-s + 0.917·19-s − 1.96·21-s + 1.45·23-s + 1.73·27-s + 1.67·29-s − 1.07·31-s + 2.08·33-s + 0.507·35-s + 0.657·37-s + 1.92·39-s − 1.40·41-s − 0.609·43-s − 0.894·45-s + 0.437·47-s + 49-s + 1.68·51-s + 1.64·53-s − 0.539·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.188735839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.188735839\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 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 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 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
−9.711676928177112821308923450659, −9.286516392215275375626881020447, −8.820509884844381138709402753007, −8.684205340604873195326364701413, −8.077001323888815233106077750829, −8.034347358198727872612474359148, −7.09647897740824745320782206073, −7.03247170236136401176869984337, −6.77301527900561703522936967243, −6.20910172218835029345130070436, −5.48529482331468642891240993658, −5.25545604249761292761246106319, −4.34169950456451886517474604303, −3.99332336757437231258778983069, −3.49435726826145050355321227370, −3.36095979141425679052665092176, −2.84595659644318422784676185891, −2.27032257874278009754569112778, −1.24681582128021905837638373273, −1.02074679397857580838969339398,
1.02074679397857580838969339398, 1.24681582128021905837638373273, 2.27032257874278009754569112778, 2.84595659644318422784676185891, 3.36095979141425679052665092176, 3.49435726826145050355321227370, 3.99332336757437231258778983069, 4.34169950456451886517474604303, 5.25545604249761292761246106319, 5.48529482331468642891240993658, 6.20910172218835029345130070436, 6.77301527900561703522936967243, 7.03247170236136401176869984337, 7.09647897740824745320782206073, 8.034347358198727872612474359148, 8.077001323888815233106077750829, 8.684205340604873195326364701413, 8.820509884844381138709402753007, 9.286516392215275375626881020447, 9.711676928177112821308923450659
