| L(s) = 1 | + 4·5-s − 4·7-s + 16·11-s − 32·13-s + 40·17-s + 56·23-s + 16·25-s + 16·31-s − 16·35-s + 64·37-s + 56·41-s + 8·43-s + 128·47-s + 8·49-s − 56·53-s + 64·55-s + 200·61-s − 128·65-s + 200·67-s − 272·71-s + 76·73-s − 64·77-s − 16·83-s + 160·85-s + 128·91-s − 20·97-s − 136·101-s + ⋯ |
| L(s) = 1 | + 4/5·5-s − 4/7·7-s + 1.45·11-s − 2.46·13-s + 2.35·17-s + 2.43·23-s + 0.639·25-s + 0.516·31-s − 0.457·35-s + 1.72·37-s + 1.36·41-s + 8/43·43-s + 2.72·47-s + 8/49·49-s − 1.05·53-s + 1.16·55-s + 3.27·61-s − 1.96·65-s + 2.98·67-s − 3.83·71-s + 1.04·73-s − 0.831·77-s − 0.192·83-s + 1.88·85-s + 1.40·91-s − 0.206·97-s − 1.34·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.184543199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.184543199\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 156 T^{3} + 2942 T^{4} + 156 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 204 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 9120 T^{3} + 148994 T^{4} + 9120 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 15240 T^{3} + 281858 T^{4} - 15240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 50904 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1722 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 64 T + 2048 T^{2} - 58176 T^{3} + 1440962 T^{4} - 58176 p^{2} T^{5} + 2048 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 28 T + 3342 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 5256 T^{3} - 557566 T^{4} - 5256 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 506496 p^{2} T^{5} + 8192 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 155064 T^{3} + 15333122 T^{4} + 155064 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1888200 T^{3} + 153742658 T^{4} - 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 68 T + p^{2} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 65436 p^{2} T^{5} + 2888 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 101328 T^{3} + 79904642 T^{4} + 101328 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 173820 T^{3} + 150551438 T^{4} + 173820 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.39089369640171370427919078246, −6.95939220796836041625517202984, −6.75164363371496396275864148867, −6.73476083580465543577626914816, −6.48591530529845118726979117616, −5.80205428290432154844952443285, −5.79207816325653523551212962519, −5.78836042259144113437244386964, −5.50691641841415878222794396133, −5.11853925612595649880833132676, −4.80518365547845531956275964306, −4.59727931034175964182317067101, −4.55092220807029211947299777248, −4.08312257004678563274434663325, −3.60779389707662713366094841090, −3.53756704873781882994952578496, −3.32118363187675438825751867826, −2.64257411825442332730202777292, −2.62672157095913647044793128373, −2.44634847105085855999083559655, −2.18135543878016811201394172803, −1.24709273617010120942840450320, −1.19256630674800865850113568032, −0.919700149875209498936954896984, −0.47763320420846956106601585202,
0.47763320420846956106601585202, 0.919700149875209498936954896984, 1.19256630674800865850113568032, 1.24709273617010120942840450320, 2.18135543878016811201394172803, 2.44634847105085855999083559655, 2.62672157095913647044793128373, 2.64257411825442332730202777292, 3.32118363187675438825751867826, 3.53756704873781882994952578496, 3.60779389707662713366094841090, 4.08312257004678563274434663325, 4.55092220807029211947299777248, 4.59727931034175964182317067101, 4.80518365547845531956275964306, 5.11853925612595649880833132676, 5.50691641841415878222794396133, 5.78836042259144113437244386964, 5.79207816325653523551212962519, 5.80205428290432154844952443285, 6.48591530529845118726979117616, 6.73476083580465543577626914816, 6.75164363371496396275864148867, 6.95939220796836041625517202984, 7.39089369640171370427919078246
