| L(s) = 1 | − 2-s + 3·3-s − 3·6-s + 3·7-s − 2·8-s + 6·9-s + 6·11-s + 6·13-s − 3·14-s + 3·16-s − 6·18-s + 6·19-s + 9·21-s − 6·22-s − 4·23-s − 6·24-s − 6·26-s + 10·27-s + 2·29-s + 2·31-s − 3·32-s + 18·33-s + 4·37-s − 6·38-s + 18·39-s + 2·41-s − 9·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1.22·6-s + 1.13·7-s − 0.707·8-s + 2·9-s + 1.80·11-s + 1.66·13-s − 0.801·14-s + 3/4·16-s − 1.41·18-s + 1.37·19-s + 1.96·21-s − 1.27·22-s − 0.834·23-s − 1.22·24-s − 1.17·26-s + 1.92·27-s + 0.371·29-s + 0.359·31-s − 0.530·32-s + 3.13·33-s + 0.657·37-s − 0.973·38-s + 2.88·39-s + 0.312·41-s − 1.38·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.154458227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.154458227\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T - 15 T^{2} - 488 T^{3} - 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 16 T + 113 T^{2} - 608 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 311 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.488955697269998902726127343570, −9.311266427577489277333430127999, −9.240701171692835591998690599415, −8.709674419243524329560312130844, −8.486807367448386961998377543363, −8.260314135259506262863829014208, −8.095944194541780680855303610191, −7.68234739127162575410322514300, −7.59127026891978329060420902585, −6.82472178482904034539484641598, −6.79374606396769027039006230987, −6.24284072893515137486339851277, −6.22924475067329073486711598694, −5.63385722833276316572021417302, −5.02522262844140395814578506502, −4.98967227666984831662205492010, −4.15816000723335585044553795505, −4.05716625139708045112469583870, −3.66481207120444030082643463708, −3.31967611255563260991031607198, −3.00963324380550823526572834362, −2.33657418047064760612703806245, −1.85573080646300267740297235574, −1.22392924948101114786523316269, −1.14951268463002916512455010277,
1.14951268463002916512455010277, 1.22392924948101114786523316269, 1.85573080646300267740297235574, 2.33657418047064760612703806245, 3.00963324380550823526572834362, 3.31967611255563260991031607198, 3.66481207120444030082643463708, 4.05716625139708045112469583870, 4.15816000723335585044553795505, 4.98967227666984831662205492010, 5.02522262844140395814578506502, 5.63385722833276316572021417302, 6.22924475067329073486711598694, 6.24284072893515137486339851277, 6.79374606396769027039006230987, 6.82472178482904034539484641598, 7.59127026891978329060420902585, 7.68234739127162575410322514300, 8.095944194541780680855303610191, 8.260314135259506262863829014208, 8.486807367448386961998377543363, 8.709674419243524329560312130844, 9.240701171692835591998690599415, 9.311266427577489277333430127999, 9.488955697269998902726127343570
