| L(s) = 1 | − 3-s + 4-s − 4·7-s − 2·9-s − 12-s + 3·13-s + 16-s + 5·19-s + 4·21-s + 25-s + 5·27-s − 4·28-s + 9·31-s − 2·36-s + 37-s − 3·39-s − 2·43-s − 48-s − 2·49-s + 3·52-s − 5·57-s + 19·61-s + 8·63-s + 64-s − 4·67-s + 3·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.51·7-s − 2/3·9-s − 0.288·12-s + 0.832·13-s + 1/4·16-s + 1.14·19-s + 0.872·21-s + 1/5·25-s + 0.962·27-s − 0.755·28-s + 1.61·31-s − 1/3·36-s + 0.164·37-s − 0.480·39-s − 0.304·43-s − 0.144·48-s − 2/7·49-s + 0.416·52-s − 0.662·57-s + 2.43·61-s + 1.00·63-s + 1/8·64-s − 0.488·67-s + 0.351·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.023139544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023139544\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 1549 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 35 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.940844535975764179210500850756, −9.672500917259841515927800438430, −9.106083201370197792553035757392, −8.348018631522322723730238028640, −8.132697585126576727851854447698, −7.18446529851754421574294641007, −6.77303363230273006012009087410, −6.24603514525929067174059421848, −5.95445255063284769076254818061, −5.28333312771084883347941402891, −4.60403552125548940095449821033, −3.47484103212811641927208698291, −3.25936599374360480077974385829, −2.39658604274002612137498331958, −0.880846910219342188804641307899,
0.880846910219342188804641307899, 2.39658604274002612137498331958, 3.25936599374360480077974385829, 3.47484103212811641927208698291, 4.60403552125548940095449821033, 5.28333312771084883347941402891, 5.95445255063284769076254818061, 6.24603514525929067174059421848, 6.77303363230273006012009087410, 7.18446529851754421574294641007, 8.132697585126576727851854447698, 8.348018631522322723730238028640, 9.106083201370197792553035757392, 9.672500917259841515927800438430, 9.940844535975764179210500850756
