| L(s) = 1 | + 2-s − 4-s − 5·5-s − 5·10-s + 2·11-s − 3·13-s + 16-s − 12·17-s + 3·19-s + 5·20-s + 2·22-s + 8·25-s − 3·26-s − 29-s + 3·31-s − 2·32-s − 12·34-s − 3·37-s + 3·38-s − 22·41-s − 3·43-s − 2·44-s − 9·47-s + 8·50-s + 3·52-s + 18·53-s − 10·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 2.23·5-s − 1.58·10-s + 0.603·11-s − 0.832·13-s + 1/4·16-s − 2.91·17-s + 0.688·19-s + 1.11·20-s + 0.426·22-s + 8/5·25-s − 0.588·26-s − 0.185·29-s + 0.538·31-s − 0.353·32-s − 2.05·34-s − 0.493·37-s + 0.486·38-s − 3.43·41-s − 0.457·43-s − 0.301·44-s − 1.31·47-s + 1.13·50-s + 0.416·52-s + 2.47·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 3 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + p T + 17 T^{2} + 39 T^{3} + 17 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 14 T^{2} + 3 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 12 T + 90 T^{2} + 435 T^{3} + 90 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 36 T^{2} + 9 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 83 T^{2} + 57 T^{3} + 83 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 213 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 22 T + 278 T^{2} + 2157 T^{3} + 278 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 63 T^{2} + 379 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 87 T^{2} + 657 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 9 T + 171 T^{2} + 999 T^{3} + 171 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 162 T^{2} - 665 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 681 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 288 T^{2} + 2019 T^{3} + 288 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 116 T^{2} + 735 T^{3} + 116 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 177 T^{2} - 21 T^{3} + 177 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.76377787061621477000513549092, −7.61727622987236709503409767785, −7.27574662841533374055707962885, −7.13606002632667339234166953540, −6.92441100112976578321844199507, −6.67839331456636373152579999452, −6.51089650139329965803616046752, −6.23531678131242638370973943648, −5.74253638104982444674529576790, −5.50890611876130950840844187885, −5.06548562220362940065025192168, −5.02398560038723280490574348793, −4.73295429878113817897898775001, −4.35091696646851235085397996437, −4.31120456375264468763613519346, −4.24416895181384737998647916299, −3.66439990382828705072696184605, −3.49582154834270131951969696871, −3.42690096330913641133670655861, −3.00931333272253712871700527157, −2.45466245427184753193412304035, −2.23945444273805631871479100518, −1.89705827130811851980395696501, −1.41485963546528499859297445987, −0.984264107350135747147727588341, 0, 0, 0,
0.984264107350135747147727588341, 1.41485963546528499859297445987, 1.89705827130811851980395696501, 2.23945444273805631871479100518, 2.45466245427184753193412304035, 3.00931333272253712871700527157, 3.42690096330913641133670655861, 3.49582154834270131951969696871, 3.66439990382828705072696184605, 4.24416895181384737998647916299, 4.31120456375264468763613519346, 4.35091696646851235085397996437, 4.73295429878113817897898775001, 5.02398560038723280490574348793, 5.06548562220362940065025192168, 5.50890611876130950840844187885, 5.74253638104982444674529576790, 6.23531678131242638370973943648, 6.51089650139329965803616046752, 6.67839331456636373152579999452, 6.92441100112976578321844199507, 7.13606002632667339234166953540, 7.27574662841533374055707962885, 7.61727622987236709503409767785, 7.76377787061621477000513549092
