| L(s) = 1 | + 2-s + 4-s − 2·5-s − 2·7-s − 2·8-s − 2·10-s − 2·11-s − 6·13-s − 2·14-s − 8·17-s − 2·20-s − 2·22-s + 6·23-s + 25-s − 6·26-s − 2·28-s − 10·29-s + 4·31-s − 4·32-s − 8·34-s + 4·35-s + 8·37-s + 4·40-s + 2·41-s + 6·43-s − 2·44-s + 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.707·8-s − 0.632·10-s − 0.603·11-s − 1.66·13-s − 0.534·14-s − 1.94·17-s − 0.447·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 1.17·26-s − 0.377·28-s − 1.85·29-s + 0.718·31-s − 0.707·32-s − 1.37·34-s + 0.676·35-s + 1.31·37-s + 0.632·40-s + 0.312·41-s + 0.914·43-s − 0.301·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8850587516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8850587516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T + 3 T^{3} - 5 T^{4} + 3 p T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 6 T^{2} - 24 T^{3} - 65 T^{4} - 24 p T^{5} - 6 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 6 T + 14 T^{2} - 24 T^{3} - 153 T^{4} - 24 p T^{5} + 14 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 30 T^{2} + 120 T^{3} + 1159 T^{4} + 120 p T^{5} + 30 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 37 T^{2} + 36 T^{3} + 1352 T^{4} + 36 p T^{5} - 37 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2 T - 66 T^{2} + 24 T^{3} + 3055 T^{4} + 24 p T^{5} - 66 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 46 T^{2} + 24 T^{3} + 3327 T^{4} + 24 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 30 T^{2} + 192 T^{3} - 845 T^{4} + 192 p T^{5} - 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 97 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 30 T^{2} - 120 T^{3} + 7519 T^{4} - 120 p T^{5} - 30 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 43 T^{2} - 258 T^{3} + 324 T^{4} - 258 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 110 T^{2} - 192 T^{3} - 325 T^{4} - 192 p T^{5} + 110 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 47 T^{2} - 816 T^{3} + 17216 T^{4} - 816 p T^{5} + 47 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.043530686282579189747071781600, −7.921296801756586812359346870499, −7.75786219646703209756476414367, −7.33781844690131969673872466953, −6.98408993641879599244853822862, −6.87567547946709116715105814583, −6.76828947489300356315658884856, −6.41195756929778225478274790703, −6.24351754004800812373306090991, −5.88505688721990551254759631656, −5.40313969744825055379971540953, −5.25274025072583409314678167052, −5.14345104924548616574015001239, −4.90691562500739980656929593506, −4.46530699105546447482071695156, −4.08005808562670108268439947050, −3.70194114765628285963908975417, −3.68591498923956038649469511012, −3.49082286345633670102895803490, −2.62236508502945595992006505586, −2.56383472050128163536341837559, −2.46931585992912679144476273361, −2.11434378698500504246132114157, −1.04785178700235456643805987958, −0.34745781495859276340737750203,
0.34745781495859276340737750203, 1.04785178700235456643805987958, 2.11434378698500504246132114157, 2.46931585992912679144476273361, 2.56383472050128163536341837559, 2.62236508502945595992006505586, 3.49082286345633670102895803490, 3.68591498923956038649469511012, 3.70194114765628285963908975417, 4.08005808562670108268439947050, 4.46530699105546447482071695156, 4.90691562500739980656929593506, 5.14345104924548616574015001239, 5.25274025072583409314678167052, 5.40313969744825055379971540953, 5.88505688721990551254759631656, 6.24351754004800812373306090991, 6.41195756929778225478274790703, 6.76828947489300356315658884856, 6.87567547946709116715105814583, 6.98408993641879599244853822862, 7.33781844690131969673872466953, 7.75786219646703209756476414367, 7.921296801756586812359346870499, 8.043530686282579189747071781600
