| L(s) = 1 | + 4·2-s + 3-s + 10·4-s + 3·5-s + 4·6-s + 20·8-s + 3·9-s + 12·10-s + 10·12-s − 8·13-s + 3·15-s + 35·16-s + 12·18-s + 30·20-s + 6·23-s + 20·24-s + 5·25-s − 32·26-s + 8·27-s + 12·30-s + 56·32-s + 30·36-s − 8·39-s + 60·40-s − 18·41-s + 9·45-s + 24·46-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 5·4-s + 1.34·5-s + 1.63·6-s + 7.07·8-s + 9-s + 3.79·10-s + 2.88·12-s − 2.21·13-s + 0.774·15-s + 35/4·16-s + 2.82·18-s + 6.70·20-s + 1.25·23-s + 4.08·24-s + 25-s − 6.27·26-s + 1.53·27-s + 2.19·30-s + 9.89·32-s + 5·36-s − 1.28·39-s + 9.48·40-s − 2.81·41-s + 1.34·45-s + 3.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(44.60381443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(44.60381443\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 324 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 2184 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 2334 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 49 T^{2} + 660 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6366 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 73 T^{2} + 2760 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 205 T^{2} + 18816 T^{4} - 205 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 18051 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 13 T + 162 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.85387289169299466045722712391, −6.30597095849735591005708759598, −6.24501269294601135994408077377, −6.21064052980526673778306437603, −5.81657513244818157842941549000, −5.67759253219960676876459194717, −5.24754836124602844190963063364, −5.21986489062575304304215038602, −4.90311536913926839657572163902, −4.69774300794860612462565641912, −4.65231129068081108763746567165, −4.52143596600982870829932916128, −4.37174184590188199655129735067, −3.60673749271220186937471130670, −3.55375140120151152226146911297, −3.34570751385865511769093004358, −3.29645867635647594803691254247, −2.72568324518726942333394489658, −2.65043453865690976503425475067, −2.32272535519652762630525380616, −2.28457894640732996837154428910, −1.63508620236438126454758752304, −1.57396666702928699457865643220, −1.39510433385679228118722824199, −0.54523352781917476416152661981,
0.54523352781917476416152661981, 1.39510433385679228118722824199, 1.57396666702928699457865643220, 1.63508620236438126454758752304, 2.28457894640732996837154428910, 2.32272535519652762630525380616, 2.65043453865690976503425475067, 2.72568324518726942333394489658, 3.29645867635647594803691254247, 3.34570751385865511769093004358, 3.55375140120151152226146911297, 3.60673749271220186937471130670, 4.37174184590188199655129735067, 4.52143596600982870829932916128, 4.65231129068081108763746567165, 4.69774300794860612462565641912, 4.90311536913926839657572163902, 5.21986489062575304304215038602, 5.24754836124602844190963063364, 5.67759253219960676876459194717, 5.81657513244818157842941549000, 6.21064052980526673778306437603, 6.24501269294601135994408077377, 6.30597095849735591005708759598, 6.85387289169299466045722712391
