| L(s) = 1 | + 3·3-s − 2·4-s + 4·5-s + 8-s + 6·9-s + 3·11-s − 6·12-s + 2·13-s + 12·15-s + 4·17-s + 4·19-s − 8·20-s + 3·24-s + 2·25-s + 10·27-s + 6·29-s + 8·31-s − 4·32-s + 9·33-s − 12·36-s − 8·37-s + 6·39-s + 4·40-s − 8·41-s − 8·43-s − 6·44-s + 24·45-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 1.78·5-s + 0.353·8-s + 2·9-s + 0.904·11-s − 1.73·12-s + 0.554·13-s + 3.09·15-s + 0.970·17-s + 0.917·19-s − 1.78·20-s + 0.612·24-s + 2/5·25-s + 1.92·27-s + 1.11·29-s + 1.43·31-s − 0.707·32-s + 1.56·33-s − 2·36-s − 1.31·37-s + 0.960·39-s + 0.632·40-s − 1.24·41-s − 1.21·43-s − 0.904·44-s + 3.57·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.81035721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.81035721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} - T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 14 T^{2} - 32 T^{3} + 14 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 28 T^{2} - 56 T^{3} + 28 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 120 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 56 T^{2} - 144 T^{3} + 56 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} - 266 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 65 T^{2} - 288 T^{3} + 65 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 120 T^{2} + 590 T^{3} + 120 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 95 T^{2} + 448 T^{3} + 95 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 101 T^{2} + 480 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 142 T^{2} - 816 T^{3} + 142 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 167 T^{2} - 1028 T^{3} + 167 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 86 T^{2} - 904 T^{3} + 86 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 16 T + 219 T^{2} - 1936 T^{3} + 219 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 64 T^{2} - 428 T^{3} + 64 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 15 T^{2} - 416 T^{3} - 15 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 68 T^{2} - 504 T^{3} + 68 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 221 T^{2} + 1768 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 205 T^{2} + 696 T^{3} + 205 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 327 T^{2} - 2880 T^{3} + 327 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 111 T^{2} + 16 T^{3} + 111 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.440397989644772278985133429135, −8.115309464806273868669255916328, −8.098669913175054535460666018481, −7.57153378145971643248515349007, −7.36022013807163531488628824057, −6.90911680735786991415459150149, −6.74618635002000132020893689462, −6.59119341728744353832883548385, −6.20454071047246781426443585209, −5.79759602624576880994992442727, −5.46240768225880472051407318515, −5.28181621823878629282287293807, −5.09765696056983483513151752259, −4.61660092151039590712923655750, −4.18266559760453320750085692690, −4.12301393352255220788027491827, −3.65665845413847059418412007617, −3.29114870511531807503472029337, −3.24696401115373763317777525918, −2.60946159557131001245877039894, −2.26498340245377359248150975724, −1.99521689827076172332998953809, −1.58778216428596393451246135794, −1.12353345687320657718825059207, −0.816779729017597855306810178953,
0.816779729017597855306810178953, 1.12353345687320657718825059207, 1.58778216428596393451246135794, 1.99521689827076172332998953809, 2.26498340245377359248150975724, 2.60946159557131001245877039894, 3.24696401115373763317777525918, 3.29114870511531807503472029337, 3.65665845413847059418412007617, 4.12301393352255220788027491827, 4.18266559760453320750085692690, 4.61660092151039590712923655750, 5.09765696056983483513151752259, 5.28181621823878629282287293807, 5.46240768225880472051407318515, 5.79759602624576880994992442727, 6.20454071047246781426443585209, 6.59119341728744353832883548385, 6.74618635002000132020893689462, 6.90911680735786991415459150149, 7.36022013807163531488628824057, 7.57153378145971643248515349007, 8.098669913175054535460666018481, 8.115309464806273868669255916328, 8.440397989644772278985133429135
