| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 4·11-s − 2·13-s + 10·19-s + 6·21-s − 3·23-s − 10·27-s + 10·31-s − 12·33-s − 2·37-s + 6·39-s + 8·41-s − 16·43-s + 4·47-s − 2·49-s − 8·53-s − 30·57-s + 10·61-s − 12·63-s − 10·67-s + 9·69-s − 8·71-s − 18·73-s − 8·77-s + 14·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 1.20·11-s − 0.554·13-s + 2.29·19-s + 1.30·21-s − 0.625·23-s − 1.92·27-s + 1.79·31-s − 2.08·33-s − 0.328·37-s + 0.960·39-s + 1.24·41-s − 2.43·43-s + 0.583·47-s − 2/7·49-s − 1.09·53-s − 3.97·57-s + 1.28·61-s − 1.51·63-s − 1.22·67-s + 1.08·69-s − 0.949·71-s − 2.10·73-s − 0.911·77-s + 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.406804849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406804849\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 6 T^{2} + 4 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 19 T^{2} - 40 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 21 T^{2} + 64 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 30 T^{2} - 18 T^{3} + 30 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 10 T + 71 T^{2} - 328 T^{3} + 71 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 66 T^{2} - 18 T^{3} + 66 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 110 T^{2} - 616 T^{3} + 110 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 48 T^{2} + 256 T^{3} + 48 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 22 T^{2} - 74 T^{3} + 22 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 16 T + 149 T^{2} + 1072 T^{3} + 149 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 91 T^{2} - 160 T^{3} + 91 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 58 T^{2} + 266 T^{3} + 58 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 156 T^{2} + 18 T^{3} + 156 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 161 T^{2} - 928 T^{3} + 161 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 218 T^{2} + 1336 T^{3} + 218 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 112 T^{2} + 554 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $D_{6}$ | \( 1 + 18 T + 153 T^{2} + 1136 T^{3} + 153 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 2116 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 232 T^{2} + 1432 T^{3} + 232 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 67 T^{2} + 556 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 20 T + 203 T^{2} - 1544 T^{3} + 203 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.98174362323570044889881554390, −6.64537206269279963816912058935, −6.59741064633581634945550988909, −6.38100131094481760597121226636, −5.94086566936277682712746730939, −5.92380183888609320885716262920, −5.80987188854446661154885520550, −5.29078441227973995718011698765, −5.09466904605026794345198319381, −5.07759117152315213829401336276, −4.60095532205306933196083328365, −4.38442564878572338036617375729, −4.34432350508758017602263502376, −3.84448019219689876184288175883, −3.58578129139409391289793983171, −3.36343128303521561620444316720, −3.11190945231982873198802801925, −2.67432596828391511103625222988, −2.61871749299982898473656769770, −1.95162090301940564751516425685, −1.59025547655775855626599130890, −1.40955959955311236775891446189, −1.05867100212817633948096638123, −0.62819373828227152848323056336, −0.30669865691687520325067517659,
0.30669865691687520325067517659, 0.62819373828227152848323056336, 1.05867100212817633948096638123, 1.40955959955311236775891446189, 1.59025547655775855626599130890, 1.95162090301940564751516425685, 2.61871749299982898473656769770, 2.67432596828391511103625222988, 3.11190945231982873198802801925, 3.36343128303521561620444316720, 3.58578129139409391289793983171, 3.84448019219689876184288175883, 4.34432350508758017602263502376, 4.38442564878572338036617375729, 4.60095532205306933196083328365, 5.07759117152315213829401336276, 5.09466904605026794345198319381, 5.29078441227973995718011698765, 5.80987188854446661154885520550, 5.92380183888609320885716262920, 5.94086566936277682712746730939, 6.38100131094481760597121226636, 6.59741064633581634945550988909, 6.64537206269279963816912058935, 6.98174362323570044889881554390
