| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 5·11-s + 3·13-s − 7·17-s + 6·19-s + 3·21-s + 23-s − 10·27-s + 10·29-s + 2·31-s − 15·33-s − 11·37-s − 9·39-s − 41-s + 10·47-s + 21·51-s − 3·53-s − 18·57-s + 8·59-s − 3·61-s − 6·63-s + 12·67-s − 3·69-s − 19·71-s − 26·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s − 1.69·17-s + 1.37·19-s + 0.654·21-s + 0.208·23-s − 1.92·27-s + 1.85·29-s + 0.359·31-s − 2.61·33-s − 1.80·37-s − 1.44·39-s − 0.156·41-s + 1.45·47-s + 2.94·51-s − 0.412·53-s − 2.38·57-s + 1.04·59-s − 0.384·61-s − 0.755·63-s + 1.46·67-s − 0.361·69-s − 2.25·71-s − 3.04·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.451284552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.451284552\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + T + T^{2} - 26 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 25 T^{2} - 78 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T - T^{2} - 118 T^{3} - p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 212 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 49 T^{2} - 6 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 75 T^{2} - 396 T^{3} + 75 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 29 T^{2} - 156 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 11 T + 135 T^{2} + 794 T^{3} + 135 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 107 T^{2} + 86 T^{3} + 107 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 256 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 109 T^{2} - 684 T^{3} + 109 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 154 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 80 T^{3} + 17 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 71 T^{2} + 202 T^{3} + 71 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 137 T^{2} - 968 T^{3} + 137 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 19 T + 317 T^{2} + 2858 T^{3} + 317 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 26 T + 399 T^{2} + 4124 T^{3} + 399 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 19 T + 341 T^{2} - 3162 T^{3} + 341 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 984 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 13 T + 307 T^{2} - 2334 T^{3} + 307 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 135 T^{2} + 326 T^{3} + 135 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.93041136517787506798242368227, −6.63871240153221342085490185068, −6.40893911343800686920452359440, −6.33926956923891183443138056979, −6.06637924896741067587609129200, −5.76087691844769913100860837969, −5.66678247383174095845090383777, −5.30008785360804659154846257976, −5.01919880369671783130541656517, −4.90291986094831428099672982859, −4.53937453136640516489088813889, −4.34240483211921496672096916757, −4.16795881891213329660893751011, −3.89840924713107628819073531767, −3.54123521895968945283985375720, −3.38732111244310785176701667101, −2.92917790231481529802574610130, −2.82491329707701183641655208417, −2.35206918202470307670813960654, −1.77866166453167689708972984679, −1.75692941219339960756662673265, −1.40455772732931851586494136309, −0.843583357669365926087712123350, −0.77465680041494378385884869220, −0.37001951311171439947614146040,
0.37001951311171439947614146040, 0.77465680041494378385884869220, 0.843583357669365926087712123350, 1.40455772732931851586494136309, 1.75692941219339960756662673265, 1.77866166453167689708972984679, 2.35206918202470307670813960654, 2.82491329707701183641655208417, 2.92917790231481529802574610130, 3.38732111244310785176701667101, 3.54123521895968945283985375720, 3.89840924713107628819073531767, 4.16795881891213329660893751011, 4.34240483211921496672096916757, 4.53937453136640516489088813889, 4.90291986094831428099672982859, 5.01919880369671783130541656517, 5.30008785360804659154846257976, 5.66678247383174095845090383777, 5.76087691844769913100860837969, 6.06637924896741067587609129200, 6.33926956923891183443138056979, 6.40893911343800686920452359440, 6.63871240153221342085490185068, 6.93041136517787506798242368227
