| L(s) = 1 | − 4·2-s − 2·3-s + 10·4-s − 2·5-s + 8·6-s − 2·7-s − 20·8-s + 9-s + 8·10-s + 2·11-s − 20·12-s − 4·13-s + 8·14-s + 4·15-s + 35·16-s + 4·17-s − 4·18-s − 4·19-s − 20·20-s + 4·21-s − 8·22-s + 8·23-s + 40·24-s + 25-s + 16·26-s + 2·27-s − 20·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.15·3-s + 5·4-s − 0.894·5-s + 3.26·6-s − 0.755·7-s − 7.07·8-s + 1/3·9-s + 2.52·10-s + 0.603·11-s − 5.77·12-s − 1.10·13-s + 2.13·14-s + 1.03·15-s + 35/4·16-s + 0.970·17-s − 0.942·18-s − 0.917·19-s − 4.47·20-s + 0.872·21-s − 1.70·22-s + 1.66·23-s + 8.16·24-s + 1/5·25-s + 3.13·26-s + 0.384·27-s − 3.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06348406633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06348406633\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 96 T^{3} - 461 T^{4} + 96 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 96 T^{3} - 469 T^{4} - 96 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + 2 T^{2} + 96 T^{3} + 107 T^{4} + 96 p T^{5} + 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 - 54 T^{2} + 1235 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 58 T^{2} + 96 T^{3} + 6107 T^{4} + 96 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 768 T^{3} - 7469 T^{4} - 768 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} + 912 T^{3} - 10941 T^{4} + 912 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 111 T^{2} - 114 T^{3} + 11732 T^{4} - 114 p T^{5} - 111 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 + 10 T + 191 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.40310846588771260934708763319, −7.04358485733164412061460734172, −6.84984285691560646460102285922, −6.72527547068686720989092953762, −6.38353107943677925950269163073, −6.32620493763640991813724636748, −6.08550798902876870187403629709, −5.69527921613358295539543162845, −5.49137040459936073345599428862, −5.32211297424046772191797192797, −4.98901597949311110759811213583, −4.75639522565398836729022964013, −4.43960625838953508121481906515, −3.96038026120998768426769521792, −3.71794807748740401632336724225, −3.47409892232129545401689466982, −3.32610337519141172966699550193, −2.76887853579925944867846487861, −2.58547262726325529931812474150, −2.28409383139327910595016288615, −1.94110801735605350955286700965, −1.46868765705065528659553402138, −0.878215642443231681910313791399, −0.852203425330420108772833614008, −0.16675286034182692367101090418,
0.16675286034182692367101090418, 0.852203425330420108772833614008, 0.878215642443231681910313791399, 1.46868765705065528659553402138, 1.94110801735605350955286700965, 2.28409383139327910595016288615, 2.58547262726325529931812474150, 2.76887853579925944867846487861, 3.32610337519141172966699550193, 3.47409892232129545401689466982, 3.71794807748740401632336724225, 3.96038026120998768426769521792, 4.43960625838953508121481906515, 4.75639522565398836729022964013, 4.98901597949311110759811213583, 5.32211297424046772191797192797, 5.49137040459936073345599428862, 5.69527921613358295539543162845, 6.08550798902876870187403629709, 6.32620493763640991813724636748, 6.38353107943677925950269163073, 6.72527547068686720989092953762, 6.84984285691560646460102285922, 7.04358485733164412061460734172, 7.40310846588771260934708763319
