| L(s) = 1 | + 4·2-s + 4·4-s + 8·5-s − 16·8-s + 32·10-s − 10·13-s − 64·16-s + 32·17-s + 32·20-s + 39·25-s − 40·26-s − 40·29-s − 64·32-s + 128·34-s + 140·37-s − 128·40-s − 160·41-s − 98·49-s + 156·50-s − 40·52-s + 224·53-s − 160·58-s − 22·61-s + 192·64-s − 80·65-s + 128·68-s − 220·73-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s + 8/5·5-s − 2·8-s + 16/5·10-s − 0.769·13-s − 4·16-s + 1.88·17-s + 8/5·20-s + 1.55·25-s − 1.53·26-s − 1.37·29-s − 2·32-s + 3.76·34-s + 3.78·37-s − 3.19·40-s − 3.90·41-s − 2·49-s + 3.11·50-s − 0.769·52-s + 4.22·53-s − 2.75·58-s − 0.360·61-s + 3·64-s − 1.23·65-s + 1.88·68-s − 3.01·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.332793854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.332793854\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2}( 1 + 8 T + 39 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2}( 1 - 10 T - 69 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T - 33 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2}( 1 - 40 T + 759 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 70 T + 3531 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 80 T + 4719 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2}( 1 - 22 T - 3237 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 110 T + 6771 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T + 17679 T^{2} - 160 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 130 T + 7491 T^{2} - 130 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.246341770485677283225294874001, −7.83856658071600178700667449976, −7.79001636856278047859563942057, −7.27763753754664579312967869494, −6.97275689997351419817989493841, −6.92261068184372270744492572113, −6.24649706932911907289252387659, −6.19474849763016507358748106701, −6.16759826537771135773891714472, −5.55764884525515908690118602326, −5.50382691549982781065233221951, −5.49616851510043383794779316513, −4.82003449637990640836671278631, −4.78472198699973980187783933167, −4.73129333913057055608779862139, −4.04207731572283003254845633768, −3.83187432568126364301512895173, −3.46191523489280002114682951692, −3.25859273943149009453919889499, −2.87918067157603476855504612500, −2.44106473290052039714036845648, −2.27873611170642453589274816899, −1.67455677294030705448451307750, −1.10573625336444939947850086521, −0.43869045333165453002045447098,
0.43869045333165453002045447098, 1.10573625336444939947850086521, 1.67455677294030705448451307750, 2.27873611170642453589274816899, 2.44106473290052039714036845648, 2.87918067157603476855504612500, 3.25859273943149009453919889499, 3.46191523489280002114682951692, 3.83187432568126364301512895173, 4.04207731572283003254845633768, 4.73129333913057055608779862139, 4.78472198699973980187783933167, 4.82003449637990640836671278631, 5.49616851510043383794779316513, 5.50382691549982781065233221951, 5.55764884525515908690118602326, 6.16759826537771135773891714472, 6.19474849763016507358748106701, 6.24649706932911907289252387659, 6.92261068184372270744492572113, 6.97275689997351419817989493841, 7.27763753754664579312967869494, 7.79001636856278047859563942057, 7.83856658071600178700667449976, 8.246341770485677283225294874001
