| L(s) = 1 | − 3·3-s − 2·4-s − 5·5-s + 3·7-s − 2·8-s + 6·9-s + 4·11-s + 6·12-s + 10·13-s + 15·15-s + 3·17-s − 19-s + 10·20-s − 9·21-s + 23-s + 6·24-s + 5·25-s − 10·27-s − 6·28-s + 4·29-s − 16·31-s + 8·32-s − 12·33-s − 15·35-s − 12·36-s − 19·37-s − 30·39-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s − 2.23·5-s + 1.13·7-s − 0.707·8-s + 2·9-s + 1.20·11-s + 1.73·12-s + 2.77·13-s + 3.87·15-s + 0.727·17-s − 0.229·19-s + 2.23·20-s − 1.96·21-s + 0.208·23-s + 1.22·24-s + 25-s − 1.92·27-s − 1.13·28-s + 0.742·29-s − 2.87·31-s + 1.41·32-s − 2.08·33-s − 2.53·35-s − 2·36-s − 3.12·37-s − 4.80·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 17^{3} \cdot 79^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 17^{3} \cdot 79^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| 79 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + p T + 4 p T^{2} + 49 T^{3} + 4 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 10 T + 67 T^{2} - 280 T^{3} + 67 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + T + 52 T^{2} + 37 T^{3} + 52 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 48 T^{2} - 75 T^{3} + 48 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 71 T^{2} - 200 T^{3} + 71 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 16 T + 157 T^{2} + 1024 T^{3} + 157 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 19 T + 218 T^{2} + 1575 T^{3} + 218 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 60 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 133 T^{2} - 20 p T^{3} + 133 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 164 T^{2} - 859 T^{3} + 164 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 17 T + 68 T^{2} + 47 T^{3} + 68 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + T + 164 T^{2} + 95 T^{3} + 164 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 17 T + 242 T^{2} + 1965 T^{3} + 242 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 149 T^{2} + 1004 T^{3} + 149 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 221 T^{2} + 1138 T^{3} + 221 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 261 T^{2} + 2054 T^{3} + 261 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 285 T^{2} - 2646 T^{3} + 285 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 4 T + 139 T^{2} + 1128 T^{3} + 139 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 281 T^{2} + 738 T^{3} + 281 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.952175864828672040747777661346, −7.50740653659740940106176097882, −7.37175633377474571129799991821, −7.13909275145762287307098906785, −6.92810216999407434916131898409, −6.53358364011601964055851854694, −6.20201726466292473034902984876, −6.17582799543902873977070063785, −5.82428178187939199254432164941, −5.62722553166650735228088724206, −5.17502540234050022118618138543, −5.15557459913576389999642639889, −4.95112563238974988669962134952, −4.24206883561560947188951861024, −4.18684094727970303192967189956, −4.14637997021622905044651021971, −3.79287620836382493931639511613, −3.69546382270552799196243784535, −3.45699692394867264586168534729, −3.00717991254265907063111670918, −2.53343308988690201442766771058, −1.83099752204562142971899098118, −1.33617199897180113315049314583, −1.26066758937245580898068466230, −1.18828196683374123072640667184, 0, 0, 0,
1.18828196683374123072640667184, 1.26066758937245580898068466230, 1.33617199897180113315049314583, 1.83099752204562142971899098118, 2.53343308988690201442766771058, 3.00717991254265907063111670918, 3.45699692394867264586168534729, 3.69546382270552799196243784535, 3.79287620836382493931639511613, 4.14637997021622905044651021971, 4.18684094727970303192967189956, 4.24206883561560947188951861024, 4.95112563238974988669962134952, 5.15557459913576389999642639889, 5.17502540234050022118618138543, 5.62722553166650735228088724206, 5.82428178187939199254432164941, 6.17582799543902873977070063785, 6.20201726466292473034902984876, 6.53358364011601964055851854694, 6.92810216999407434916131898409, 7.13909275145762287307098906785, 7.37175633377474571129799991821, 7.50740653659740940106176097882, 7.952175864828672040747777661346
