| L(s) = 1 | + 2·3-s + 2·5-s + 3·7-s + 9-s − 2·11-s + 3·13-s + 4·15-s + 4·17-s + 4·19-s + 6·21-s − 10·23-s − 8·25-s + 24·29-s + 4·31-s − 4·33-s + 6·35-s + 6·39-s + 2·41-s − 10·43-s + 2·45-s + 8·47-s + 6·49-s + 8·51-s + 8·53-s − 4·55-s + 8·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.832·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 1.30·21-s − 2.08·23-s − 8/5·25-s + 4.45·29-s + 0.718·31-s − 0.696·33-s + 1.01·35-s + 0.960·39-s + 0.312·41-s − 1.52·43-s + 0.298·45-s + 1.16·47-s + 6/7·49-s + 1.12·51-s + 1.09·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.576982898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.576982898\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 12 T^{2} - 18 T^{3} + 12 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 36 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 58 T^{2} - 148 T^{3} + 58 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 70 T^{2} + 324 T^{3} + 70 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 74 T^{2} - 264 T^{3} + 74 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 95 T^{2} - 172 T^{3} + 95 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 58 T^{2} + 232 T^{3} + 58 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 62 T^{2} - 208 T^{3} + 62 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 124 T^{2} - 870 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 216 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 868 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 120 T^{2} + 1186 T^{3} + 120 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 242 T^{2} - 2196 T^{3} + 242 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 172 T^{2} + 66 T^{3} + 172 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 320 T^{2} + 1962 T^{3} + 320 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.547364917649146820897717181080, −8.243895731884702410457988735007, −7.946585398528833153721002390058, −7.70796352454159055241506083562, −7.64357321295259739145065555352, −7.29783290257583953046814290682, −6.70802981775855558616475621121, −6.34009340493963960645564380591, −6.29115308760223016850462173196, −6.16142811753794880321520619207, −5.42230861252020864753043610279, −5.37184436930791769963718952331, −5.30493396736537125383925114915, −4.65725148729726614126099270986, −4.33017221439190916384393138547, −4.29190655729097811687734341500, −3.68059238639301667916185921862, −3.31836996127260909596993916551, −3.19442215475345536716127725010, −2.61236956096011122294262688049, −2.31185673482776118411148128145, −2.11654988540624096334527919707, −1.67601087289506257611219709057, −1.01498067963245274444028235850, −0.822900904814127900151644335214,
0.822900904814127900151644335214, 1.01498067963245274444028235850, 1.67601087289506257611219709057, 2.11654988540624096334527919707, 2.31185673482776118411148128145, 2.61236956096011122294262688049, 3.19442215475345536716127725010, 3.31836996127260909596993916551, 3.68059238639301667916185921862, 4.29190655729097811687734341500, 4.33017221439190916384393138547, 4.65725148729726614126099270986, 5.30493396736537125383925114915, 5.37184436930791769963718952331, 5.42230861252020864753043610279, 6.16142811753794880321520619207, 6.29115308760223016850462173196, 6.34009340493963960645564380591, 6.70802981775855558616475621121, 7.29783290257583953046814290682, 7.64357321295259739145065555352, 7.70796352454159055241506083562, 7.946585398528833153721002390058, 8.243895731884702410457988735007, 8.547364917649146820897717181080
