| L(s) = 1 | − 4·2-s + 10·4-s − 4·5-s + 5·7-s − 20·8-s + 16·10-s − 3·11-s + 6·13-s − 20·14-s + 35·16-s + 7·19-s − 40·20-s + 12·22-s − 23-s + 10·25-s − 24·26-s + 50·28-s − 4·29-s + 4·31-s − 56·32-s − 20·35-s + 6·37-s − 28·38-s + 80·40-s − 4·41-s + 13·43-s − 30·44-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 1.78·5-s + 1.88·7-s − 7.07·8-s + 5.05·10-s − 0.904·11-s + 1.66·13-s − 5.34·14-s + 35/4·16-s + 1.60·19-s − 8.94·20-s + 2.55·22-s − 0.208·23-s + 2·25-s − 4.70·26-s + 9.44·28-s − 0.742·29-s + 0.718·31-s − 9.89·32-s − 3.38·35-s + 0.986·37-s − 4.54·38-s + 12.6·40-s − 0.624·41-s + 1.98·43-s − 4.52·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610861430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610861430\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 5 T + 24 T^{2} - 81 T^{3} + 254 T^{4} - 81 p T^{5} + 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 3 T + 34 T^{2} + 79 T^{3} + 530 T^{4} + 79 p T^{5} + 34 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 6 T + 40 T^{2} - 178 T^{3} + 766 T^{4} - 178 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 28 T^{2} + 80 T^{3} + 422 T^{4} + 80 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 7 T + 64 T^{2} - 351 T^{3} + 1774 T^{4} - 351 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + T + 56 T^{2} + 149 T^{3} + 1470 T^{4} + 149 p T^{5} + 56 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 4 T + 20 T^{2} + 92 T^{3} + 1014 T^{4} + 92 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 48 T^{2} - 290 T^{3} + 3086 T^{4} - 290 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 68 T^{2} + 236 T^{3} + 3750 T^{4} + 236 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 13 T + 104 T^{2} - 925 T^{3} + 7870 T^{4} - 925 p T^{5} + 104 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 4 T + 76 T^{2} - 244 T^{3} + 4262 T^{4} - 244 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 174 T^{2} - 727 T^{3} + 12802 T^{4} - 727 p T^{5} + 174 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 8 T + 84 T^{2} + 568 T^{3} + 6022 T^{4} + 568 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 4 T - 12 T^{2} + 348 T^{3} + 6470 T^{4} + 348 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 8 T + 108 T^{2} + 360 T^{3} - 58 T^{4} + 360 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - T + 186 T^{2} - 185 T^{3} + 18466 T^{4} - 185 p T^{5} + 186 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 15 T + 256 T^{2} - 2149 T^{3} + 23710 T^{4} - 2149 p T^{5} + 256 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 5 T + 160 T^{2} - 1393 T^{3} + 12734 T^{4} - 1393 p T^{5} + 160 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2 T + 188 T^{2} - 1138 T^{3} + 16662 T^{4} - 1138 p T^{5} + 188 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 9 T + 118 T^{2} + 329 T^{3} + 1394 T^{4} + 329 p T^{5} + 118 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 10 T + 304 T^{2} - 2070 T^{3} + 40030 T^{4} - 2070 p T^{5} + 304 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.44967253537457146027898926534, −6.03759429125677744679805746085, −5.99106178563630460917042034188, −5.71147320309174252185903230588, −5.59214028612187561373555415873, −5.09128807120580688592606078859, −5.04532554680886401281552173168, −4.92684752175251837113606763455, −4.72444519509927448396174134588, −4.15925891827925806725940895572, −4.01152825054695068992193717052, −3.88841394475031871037839270037, −3.85807640909542944040320047866, −3.06328348461637608444427343297, −3.04899269269836024182274431269, −2.98607088277888542708005470902, −2.97934026203930142165974950223, −2.11803709982847350975517260360, −1.98255781370053761107181410394, −1.83987922280783240194920090603, −1.72780972964094610765890792840, −1.00507496397471596218587421216, −0.888767666197895784532378805436, −0.67101706236641909982847929255, −0.52296432586472041904313330071,
0.52296432586472041904313330071, 0.67101706236641909982847929255, 0.888767666197895784532378805436, 1.00507496397471596218587421216, 1.72780972964094610765890792840, 1.83987922280783240194920090603, 1.98255781370053761107181410394, 2.11803709982847350975517260360, 2.97934026203930142165974950223, 2.98607088277888542708005470902, 3.04899269269836024182274431269, 3.06328348461637608444427343297, 3.85807640909542944040320047866, 3.88841394475031871037839270037, 4.01152825054695068992193717052, 4.15925891827925806725940895572, 4.72444519509927448396174134588, 4.92684752175251837113606763455, 5.04532554680886401281552173168, 5.09128807120580688592606078859, 5.59214028612187561373555415873, 5.71147320309174252185903230588, 5.99106178563630460917042034188, 6.03759429125677744679805746085, 6.44967253537457146027898926534
