| L(s) = 1 | + 3·3-s + 6·5-s + 4·7-s + 6·9-s + 4·11-s + 6·13-s + 18·15-s − 4·19-s + 12·21-s + 24·23-s + 13·25-s + 10·27-s − 14·29-s + 4·31-s + 12·33-s + 24·35-s − 10·37-s + 18·39-s + 12·41-s + 8·43-s + 36·45-s + 3·49-s − 8·53-s + 24·55-s − 12·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.68·5-s + 1.51·7-s + 2·9-s + 1.20·11-s + 1.66·13-s + 4.64·15-s − 0.917·19-s + 2.61·21-s + 5.00·23-s + 13/5·25-s + 1.92·27-s − 2.59·29-s + 0.718·31-s + 2.08·33-s + 4.05·35-s − 1.64·37-s + 2.88·39-s + 1.87·41-s + 1.21·43-s + 5.36·45-s + 3/7·49-s − 1.09·53-s + 3.23·55-s − 1.58·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(53.76323575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(53.76323575\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 167 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 6 T + 23 T^{2} - 62 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 40 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 29 T^{2} - 68 T^{3} + 29 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 15 T^{2} + 54 T^{3} + 15 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 120 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 + 14 T + 115 T^{2} + 660 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 77 T^{2} - 216 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 131 T^{2} + 732 T^{3} + 131 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 51 T^{2} - 66 T^{3} + 51 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 113 T^{2} - 558 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 57 T^{2} + 268 T^{3} + 57 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 115 T^{2} + 558 T^{3} + 115 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 145 T^{2} + 504 T^{3} + 145 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 248 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 26 T + 413 T^{2} - 4038 T^{3} + 413 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 149 T^{2} - 1232 T^{3} + 149 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 319 T^{2} - 3316 T^{3} + 319 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 10 T + 137 T^{2} + 1302 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T - 7 T^{2} + 760 T^{3} - 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 148 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 159 T^{2} - 1316 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85315062006866494661823473901, −6.63127350501877074041447868908, −6.49987320077356781486333119939, −6.49728512445957312250585750867, −5.92762422296789811980698165848, −5.66384670890274165497363500711, −5.53414944520892194423321004693, −5.27209331414096875043652863469, −5.04482415157639593257895431197, −4.94824005482558571566438195811, −4.41931166333082150945384555645, −4.37020740599241993160347859143, −3.94981684215197929446527605346, −3.55774403277155744355347226955, −3.46612934121373470598499432855, −3.42133564502810411654575998935, −2.73654434619133744962715104978, −2.61328585197223690846052515925, −2.35690546945413262201974087124, −1.93595892610575022875278059804, −1.79651353102340875069989990423, −1.73110281227054776356746292803, −1.11684699911834793097319554518, −1.03638863439279505037005614229, −0.880743890051077503912602771728,
0.880743890051077503912602771728, 1.03638863439279505037005614229, 1.11684699911834793097319554518, 1.73110281227054776356746292803, 1.79651353102340875069989990423, 1.93595892610575022875278059804, 2.35690546945413262201974087124, 2.61328585197223690846052515925, 2.73654434619133744962715104978, 3.42133564502810411654575998935, 3.46612934121373470598499432855, 3.55774403277155744355347226955, 3.94981684215197929446527605346, 4.37020740599241993160347859143, 4.41931166333082150945384555645, 4.94824005482558571566438195811, 5.04482415157639593257895431197, 5.27209331414096875043652863469, 5.53414944520892194423321004693, 5.66384670890274165497363500711, 5.92762422296789811980698165848, 6.49728512445957312250585750867, 6.49987320077356781486333119939, 6.63127350501877074041447868908, 6.85315062006866494661823473901
