| L(s) = 1 | + 3-s + 3·5-s − 7-s + 3·15-s − 7·17-s + 3·19-s − 21-s + 4·23-s + 6·25-s + 27-s − 5·29-s + 17·31-s − 3·35-s + 37-s + 2·41-s + 16·43-s + 6·47-s − 12·49-s − 7·51-s + 53-s + 3·57-s + 4·59-s − 11·61-s + 16·67-s + 4·69-s − 71-s + 22·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 0.774·15-s − 1.69·17-s + 0.688·19-s − 0.218·21-s + 0.834·23-s + 6/5·25-s + 0.192·27-s − 0.928·29-s + 3.05·31-s − 0.507·35-s + 0.164·37-s + 0.312·41-s + 2.43·43-s + 0.875·47-s − 1.71·49-s − 0.980·51-s + 0.137·53-s + 0.397·57-s + 0.520·59-s − 1.40·61-s + 1.95·67-s + 0.481·69-s − 0.118·71-s + 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.92144210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.92144210\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + T^{2} - 2 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} + 10 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T^{2} - 64 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 59 T^{2} + 230 T^{3} + 59 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 17 T^{2} - 130 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 71 T^{2} + 226 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 165 T^{2} - 1070 T^{3} + 165 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - T + 87 T^{2} - 94 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 91 T^{2} - 132 T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 189 T^{2} - 1360 T^{3} + 189 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 113 T^{2} - 556 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 135 T^{2} - 126 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 81 T^{2} - 600 T^{3} + 81 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 11 T + 95 T^{2} + 6 p T^{3} + 95 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 165 T^{2} - 1168 T^{3} + 165 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + T + 85 T^{2} + 654 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 355 T^{2} - 3388 T^{3} + 355 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 880 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 83 T^{2} + 1138 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
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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
−6.81536576017082901890951337774, −6.43364966222057861442194262641, −6.37546742156865966344056596767, −6.18264682903485965925015072763, −5.94311100845211582892545258324, −5.63778862520529089286875274799, −5.51064038249860586115086560871, −5.02041074488077072463101863985, −4.93426243357501218821942298246, −4.66605153796993040813196465347, −4.58978218514600644447283448560, −4.25326122194598661344793578469, −3.92739205104372921978307207270, −3.49091796461213655051491995745, −3.33206338406025818099414574435, −3.31262627352069103180232129545, −2.70940096289806897509148276314, −2.48867473817428016200642733849, −2.37912396688920701217631372703, −2.13366419772419433811967890920, −1.96725921779954477840261696818, −1.37158493718043754183518143793, −0.976010292333823414274707367195, −0.76643553671172750274623175089, −0.50923874372048305079146366620,
0.50923874372048305079146366620, 0.76643553671172750274623175089, 0.976010292333823414274707367195, 1.37158493718043754183518143793, 1.96725921779954477840261696818, 2.13366419772419433811967890920, 2.37912396688920701217631372703, 2.48867473817428016200642733849, 2.70940096289806897509148276314, 3.31262627352069103180232129545, 3.33206338406025818099414574435, 3.49091796461213655051491995745, 3.92739205104372921978307207270, 4.25326122194598661344793578469, 4.58978218514600644447283448560, 4.66605153796993040813196465347, 4.93426243357501218821942298246, 5.02041074488077072463101863985, 5.51064038249860586115086560871, 5.63778862520529089286875274799, 5.94311100845211582892545258324, 6.18264682903485965925015072763, 6.37546742156865966344056596767, 6.43364966222057861442194262641, 6.81536576017082901890951337774
