| L(s) = 1 | + 3·2-s + 6·4-s − 5-s + 10·8-s − 3·10-s − 11-s + 8·13-s + 15·16-s + 4·17-s − 3·19-s − 6·20-s − 3·22-s − 7·23-s − 8·25-s + 24·26-s − 5·29-s + 20·31-s + 21·32-s + 12·34-s − 3·37-s − 9·38-s − 10·40-s + 6·43-s − 6·44-s − 21·46-s + 9·47-s − 24·50-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 0.447·5-s + 3.53·8-s − 0.948·10-s − 0.301·11-s + 2.21·13-s + 15/4·16-s + 0.970·17-s − 0.688·19-s − 1.34·20-s − 0.639·22-s − 1.45·23-s − 8/5·25-s + 4.70·26-s − 0.928·29-s + 3.59·31-s + 3.71·32-s + 2.05·34-s − 0.493·37-s − 1.45·38-s − 1.58·40-s + 0.914·43-s − 0.904·44-s − 3.09·46-s + 1.31·47-s − 3.39·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(32.75527110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(32.75527110\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 13 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 40 T^{2} - 139 T^{3} + 40 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 39 T^{2} - 112 T^{3} + 39 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 81 T^{2} + 325 T^{3} + 81 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 21 T^{2} - 73 T^{3} + 21 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 20 T + 214 T^{2} - 1441 T^{3} + 214 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 90 T^{2} + 9 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 43 | $D_{6}$ | \( 1 - 6 T + 60 T^{2} - 389 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 15 T + 225 T^{2} - 1671 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 216 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 178 T^{2} - 883 T^{3} + 178 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 89 T^{2} - 77 T^{3} + 89 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 19 T + 227 T^{2} - 2143 T^{3} + 227 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 5 T + 163 T^{2} + 469 T^{3} + 163 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 186 T^{2} + 185 T^{3} + 186 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 225 T^{2} - 1611 T^{3} + 225 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 503 T^{2} - 5680 T^{3} + 503 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.81924148953008984967112266524, −6.33740288308311858933409534876, −6.32462031773942956104074743655, −6.29778021666941377537658206727, −5.85018927830208220303362116432, −5.78351694833849275374795947064, −5.69505258942688516612573812127, −5.12457526773139723693482644717, −5.10667756409909393017682259675, −4.87712409142002815459354333580, −4.23526429598147703484025508047, −4.19706864257875474098804322510, −4.19036564464024940490272591818, −3.72722318764231437928223296079, −3.64998068092732394163873289486, −3.55505180631878830641052110990, −3.05108018833996235682809206254, −2.74388257622433649285428951609, −2.56992013609269888269433201346, −2.10505788702511425628219423839, −1.88160986687871992399170087813, −1.84705407200841790236038005094, −1.01468622779629239786804554514, −0.808959057657749111890277266585, −0.64539924975442040367559036892,
0.64539924975442040367559036892, 0.808959057657749111890277266585, 1.01468622779629239786804554514, 1.84705407200841790236038005094, 1.88160986687871992399170087813, 2.10505788702511425628219423839, 2.56992013609269888269433201346, 2.74388257622433649285428951609, 3.05108018833996235682809206254, 3.55505180631878830641052110990, 3.64998068092732394163873289486, 3.72722318764231437928223296079, 4.19036564464024940490272591818, 4.19706864257875474098804322510, 4.23526429598147703484025508047, 4.87712409142002815459354333580, 5.10667756409909393017682259675, 5.12457526773139723693482644717, 5.69505258942688516612573812127, 5.78351694833849275374795947064, 5.85018927830208220303362116432, 6.29778021666941377537658206727, 6.32462031773942956104074743655, 6.33740288308311858933409534876, 6.81924148953008984967112266524
