| L(s) = 1 | − 2·4-s − 2·5-s + 4·16-s − 8·19-s + 4·20-s + 12·23-s + 3·25-s − 12·29-s + 16·43-s − 4·49-s − 24·53-s − 8·64-s + 16·67-s + 28·73-s + 16·76-s − 8·80-s − 24·92-s + 16·95-s − 20·97-s − 6·100-s − 12·101-s − 24·115-s + 24·116-s + 20·121-s − 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s + 16-s − 1.83·19-s + 0.894·20-s + 2.50·23-s + 3/5·25-s − 2.22·29-s + 2.43·43-s − 4/7·49-s − 3.29·53-s − 64-s + 1.95·67-s + 3.27·73-s + 1.83·76-s − 0.894·80-s − 2.50·92-s + 1.64·95-s − 2.03·97-s − 3/5·100-s − 1.19·101-s − 2.23·115-s + 2.22·116-s + 1.81·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7932028497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7932028497\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.45551312334139437742251812804, −11.00794501908516986370027919969, −10.97900678798030094794948149891, −10.50444401550395909437218691518, −9.437233547529379136720938173399, −9.365369084546064745117785379459, −9.122515437813939640607469354273, −8.189910695495045566734064790574, −8.141887114827673322842813735430, −7.60916422639482856644247854106, −6.72331570197606475065093664952, −6.71728405738356562238798562809, −5.60223145627249709613493317736, −5.34695776874268799749791666621, −4.46014190788157572885678232061, −4.33641379098073411985030648919, −3.54020984178657921228477008539, −3.03967244986734624438528594222, −1.92935022128585004072570941410, −0.63113860018792089149607210642,
0.63113860018792089149607210642, 1.92935022128585004072570941410, 3.03967244986734624438528594222, 3.54020984178657921228477008539, 4.33641379098073411985030648919, 4.46014190788157572885678232061, 5.34695776874268799749791666621, 5.60223145627249709613493317736, 6.71728405738356562238798562809, 6.72331570197606475065093664952, 7.60916422639482856644247854106, 8.141887114827673322842813735430, 8.189910695495045566734064790574, 9.122515437813939640607469354273, 9.365369084546064745117785379459, 9.437233547529379136720938173399, 10.50444401550395909437218691518, 10.97900678798030094794948149891, 11.00794501908516986370027919969, 11.45551312334139437742251812804
