| L(s) = 1 | + 3·3-s + 4-s − 7-s − 2·8-s + 6·9-s + 11-s + 3·12-s − 3·13-s + 3·16-s + 17-s + 6·19-s − 3·21-s + 7·23-s − 6·24-s + 10·27-s − 28-s + 18·29-s + 6·31-s − 4·32-s + 3·33-s + 6·36-s − 13·37-s − 9·39-s + 41-s + 44-s + 18·47-s + 9·48-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1/2·4-s − 0.377·7-s − 0.707·8-s + 2·9-s + 0.301·11-s + 0.866·12-s − 0.832·13-s + 3/4·16-s + 0.242·17-s + 1.37·19-s − 0.654·21-s + 1.45·23-s − 1.22·24-s + 1.92·27-s − 0.188·28-s + 3.34·29-s + 1.07·31-s − 0.707·32-s + 0.522·33-s + 36-s − 2.13·37-s − 1.44·39-s + 0.156·41-s + 0.150·44-s + 2.62·47-s + 1.29·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.312828944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.312828944\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - T^{2} + p T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 42 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 164 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 53 T^{2} - 194 T^{3} + 53 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 646 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 91 T^{2} - 6 T^{3} + 91 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 128 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 18 T + 221 T^{2} - 1756 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 167 T^{2} + 1162 T^{3} + 167 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 129 T^{2} + 816 T^{3} + 129 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 254 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 408 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 237 T^{2} + 1530 T^{3} + 237 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 119 T^{2} - 532 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 5 T + 189 T^{2} - 854 T^{3} + 189 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 201 T^{2} + 1200 T^{3} + 201 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 11 T + 275 T^{2} - 1954 T^{3} + 275 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 25 T + 467 T^{2} - 5094 T^{3} + 467 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.978178205147990256913285490661, −8.686834208183750458505261644683, −8.378399001020672648982894450179, −8.097448163349804344522488895387, −7.84261096176926237973580588508, −7.40341131345381536583039688956, −7.23346096847671458642113377924, −7.14070728287996220853862685718, −6.60869261024382651076247522709, −6.46994556445513384007715817161, −6.12145077375085764124280819691, −5.73621443665805365788928110018, −5.35950812860124869186381001083, −4.82334118053611320062057411602, −4.72031648974748641580814814830, −4.53489090238216368852424613228, −3.79404807817002040399353500790, −3.49062932289989177217503348973, −3.08005417038692581858456229087, −3.02158224611128967607886093038, −2.84053070422656402374494518071, −2.19541014199379146725501136288, −1.90145730813216072519372550098, −1.00003394295883133642957681304, −0.953026695536530234843631952437,
0.953026695536530234843631952437, 1.00003394295883133642957681304, 1.90145730813216072519372550098, 2.19541014199379146725501136288, 2.84053070422656402374494518071, 3.02158224611128967607886093038, 3.08005417038692581858456229087, 3.49062932289989177217503348973, 3.79404807817002040399353500790, 4.53489090238216368852424613228, 4.72031648974748641580814814830, 4.82334118053611320062057411602, 5.35950812860124869186381001083, 5.73621443665805365788928110018, 6.12145077375085764124280819691, 6.46994556445513384007715817161, 6.60869261024382651076247522709, 7.14070728287996220853862685718, 7.23346096847671458642113377924, 7.40341131345381536583039688956, 7.84261096176926237973580588508, 8.097448163349804344522488895387, 8.378399001020672648982894450179, 8.686834208183750458505261644683, 8.978178205147990256913285490661
