| L(s) = 1 | − 2·3-s + 4·5-s + 4·7-s + 9-s + 6·11-s − 4·13-s − 8·15-s − 4·17-s + 3·19-s − 8·21-s + 12·23-s − 2·29-s − 2·31-s − 12·33-s + 16·35-s + 2·37-s + 8·39-s − 4·41-s + 2·43-s + 4·45-s − 6·49-s + 8·51-s − 14·53-s + 24·55-s − 6·57-s − 28·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 2.06·15-s − 0.970·17-s + 0.688·19-s − 1.74·21-s + 2.50·23-s − 0.371·29-s − 0.359·31-s − 2.08·33-s + 2.70·35-s + 0.328·37-s + 1.28·39-s − 0.624·41-s + 0.304·43-s + 0.596·45-s − 6/7·49-s + 1.12·51-s − 1.92·53-s + 3.23·55-s − 0.794·57-s − 3.64·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.608381666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.608381666\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| 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 | $S_4\times C_2$ | \( 1 - 4 T + 16 T^{2} - 36 T^{3} + 16 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 22 T^{2} - 52 T^{3} + 22 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} - 128 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} + 40 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 40 T^{2} + 138 T^{3} + 40 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} - 60 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T - T^{2} - 228 T^{3} - p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 97 T^{2} - 116 T^{3} + 97 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 396 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 98 T^{2} - 216 T^{3} + 98 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 50 T^{2} - 332 T^{3} + 50 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 145 T^{2} + 1116 T^{3} + 145 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 28 T + 421 T^{2} + 3960 T^{3} + 421 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 108 T^{2} - 380 T^{3} + 108 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} - 1132 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 201 T^{2} - 1536 T^{3} + 201 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 348 T^{2} - 3458 T^{3} + 348 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 964 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 285 T^{2} - 3336 T^{3} + 285 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 209 T^{2} + 124 T^{3} + 209 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 435 T^{2} + 4540 T^{3} + 435 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.35719511755386624595130088137, −6.77189384210069846586948829692, −6.74468942644076068357999528978, −6.49265005880980053383970445727, −6.38770128856772663099748003713, −6.28653764734838695476684019692, −5.81597964795977790360781469027, −5.32374922146645445617025145365, −5.31582551738016093528200032709, −5.20269675718174215442596639782, −5.06710568154760620455916260913, −4.73595321763118076916173034678, −4.38895648021525968104682697387, −4.14309659716836915824765039441, −3.83698035257261304779477465552, −3.53430141241776225989830718458, −3.01340046928629031821041997390, −2.85498665294114843081484062362, −2.51215465402937666317429491034, −1.97767140162515434318988160592, −1.76413894623610478480351025981, −1.64539378480142204882627476698, −1.33611197125935400275519490369, −0.868825685530649333591937128766, −0.35461654319258823201497795522,
0.35461654319258823201497795522, 0.868825685530649333591937128766, 1.33611197125935400275519490369, 1.64539378480142204882627476698, 1.76413894623610478480351025981, 1.97767140162515434318988160592, 2.51215465402937666317429491034, 2.85498665294114843081484062362, 3.01340046928629031821041997390, 3.53430141241776225989830718458, 3.83698035257261304779477465552, 4.14309659716836915824765039441, 4.38895648021525968104682697387, 4.73595321763118076916173034678, 5.06710568154760620455916260913, 5.20269675718174215442596639782, 5.31582551738016093528200032709, 5.32374922146645445617025145365, 5.81597964795977790360781469027, 6.28653764734838695476684019692, 6.38770128856772663099748003713, 6.49265005880980053383970445727, 6.74468942644076068357999528978, 6.77189384210069846586948829692, 7.35719511755386624595130088137
