| L(s) = 1 | − 2·5-s + 14·17-s + 11·25-s + 5·49-s − 30·61-s − 22·73-s − 28·85-s + 40·101-s − 3·121-s − 38·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.39·17-s + 11/5·25-s + 5/7·49-s − 3.84·61-s − 2.57·73-s − 3.03·85-s + 3.98·101-s − 0.272·121-s − 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.803383140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.803383140\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} )( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.13604393540101163221459722061, −6.13535881065668641361519627083, −5.84513974566926806013730997380, −5.67565317516073225597428295887, −5.36295418401984817467550898969, −5.32520059819713431039158272755, −4.97161909276522138717738041736, −4.81735904721928808625906052292, −4.52415439635498938938854221339, −4.40981939200478667690986524616, −4.35682705620146562975115692051, −3.67587612221259471276879034555, −3.65133520122593812866803010096, −3.63064630154108936922242544463, −3.19921106142275687498963307861, −2.89791288848788442322312227279, −2.89330830437438205743543173680, −2.88442811725028228128903644658, −2.21923774169182495897521653860, −1.87598233346154234226327183218, −1.66607707720509697197046227194, −1.23029007753519793489287854890, −0.974372968958819450486375148832, −0.909996858983702918857335973493, −0.22315468211531304619483423828,
0.22315468211531304619483423828, 0.909996858983702918857335973493, 0.974372968958819450486375148832, 1.23029007753519793489287854890, 1.66607707720509697197046227194, 1.87598233346154234226327183218, 2.21923774169182495897521653860, 2.88442811725028228128903644658, 2.89330830437438205743543173680, 2.89791288848788442322312227279, 3.19921106142275687498963307861, 3.63064630154108936922242544463, 3.65133520122593812866803010096, 3.67587612221259471276879034555, 4.35682705620146562975115692051, 4.40981939200478667690986524616, 4.52415439635498938938854221339, 4.81735904721928808625906052292, 4.97161909276522138717738041736, 5.32520059819713431039158272755, 5.36295418401984817467550898969, 5.67565317516073225597428295887, 5.84513974566926806013730997380, 6.13535881065668641361519627083, 6.13604393540101163221459722061
