| L(s) = 1 | + 4·5-s + 4·7-s − 8·13-s + 12·29-s + 12·31-s + 16·35-s − 16·37-s − 4·49-s + 20·53-s − 16·61-s − 32·65-s + 16·67-s + 8·71-s − 8·73-s + 12·79-s + 8·89-s − 32·91-s + 20·101-s + 4·103-s − 24·109-s + 8·113-s − 20·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s − 2.21·13-s + 2.22·29-s + 2.15·31-s + 2.70·35-s − 2.63·37-s − 4/7·49-s + 2.74·53-s − 2.04·61-s − 3.96·65-s + 1.95·67-s + 0.949·71-s − 0.936·73-s + 1.35·79-s + 0.847·89-s − 3.35·91-s + 1.99·101-s + 0.394·103-s − 2.29·109-s + 0.752·113-s − 1.81·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.18965340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.18965340\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 16 T^{2} - 44 T^{3} + 118 T^{4} - 44 p T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 20 T^{2} - 60 T^{3} + 186 T^{4} - 60 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T^{2} + 32 T^{3} + 230 T^{4} + 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 56 T^{2} + 264 T^{3} + 1122 T^{4} + 264 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 44 T^{2} - 64 T^{3} + 966 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 144 T^{2} - 964 T^{3} + 6422 T^{4} - 964 p T^{5} + 144 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 200 T^{2} + 1552 T^{3} + 11010 T^{4} + 1552 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 100 T^{2} - 192 T^{3} + 4726 T^{4} - 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 76 T^{2} - 256 T^{3} + 2726 T^{4} - 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 336 T^{2} - 3452 T^{3} + 30134 T^{4} - 3452 p T^{5} + 336 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 296 T^{2} + 2704 T^{3} + 27618 T^{4} + 2704 p T^{5} + 296 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 4 p T^{2} - 2960 T^{3} + 27190 T^{4} - 2960 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 252 T^{2} - 1512 T^{3} + 25766 T^{4} - 1512 p T^{5} + 252 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 116 T^{2} + 160 T^{3} + 5510 T^{4} + 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 156 T^{2} - 504 T^{3} + 10022 T^{4} - 504 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.39243283953204492780182737324, −5.15871679221055202600701699607, −5.01011804036323856153603482135, −4.93482217261117851180757813543, −4.91877062813087041637966123047, −4.39338404251876167510254103809, −4.35111986509268644094144722114, −4.31814712565766639849146397174, −4.16909348135625314875352381348, −3.71718489341468301002549159158, −3.39537177498890563300372613469, −3.27363032643052051363211392454, −3.20576639931142461379748704250, −2.77420661818527796200533177698, −2.60555421751559203659579691778, −2.42992764600780078171011300336, −2.40533845182304564545371880244, −1.91696745507428839281951573818, −1.83795707156280769139226829736, −1.70427575305255566470085961311, −1.69145531827896506059156077095, −1.05981164497969476312511348271, −0.874041656161496072794554406245, −0.61663973554084814704393437465, −0.32141600898816960907333942012,
0.32141600898816960907333942012, 0.61663973554084814704393437465, 0.874041656161496072794554406245, 1.05981164497969476312511348271, 1.69145531827896506059156077095, 1.70427575305255566470085961311, 1.83795707156280769139226829736, 1.91696745507428839281951573818, 2.40533845182304564545371880244, 2.42992764600780078171011300336, 2.60555421751559203659579691778, 2.77420661818527796200533177698, 3.20576639931142461379748704250, 3.27363032643052051363211392454, 3.39537177498890563300372613469, 3.71718489341468301002549159158, 4.16909348135625314875352381348, 4.31814712565766639849146397174, 4.35111986509268644094144722114, 4.39338404251876167510254103809, 4.91877062813087041637966123047, 4.93482217261117851180757813543, 5.01011804036323856153603482135, 5.15871679221055202600701699607, 5.39243283953204492780182737324
