| L(s) = 1 | − 2·5-s + 7-s − 3·11-s − 5·17-s − 2·19-s − 7·23-s − 3·25-s − 7·29-s − 4·31-s − 2·35-s + 3·37-s − 9·41-s − 5·43-s − 7·47-s − 13·49-s − 16·53-s + 6·55-s − 17·59-s + 20·61-s − 20·67-s + 4·71-s + 12·73-s − 3·77-s + 5·79-s − 8·83-s + 10·85-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.904·11-s − 1.21·17-s − 0.458·19-s − 1.45·23-s − 3/5·25-s − 1.29·29-s − 0.718·31-s − 0.338·35-s + 0.493·37-s − 1.40·41-s − 0.762·43-s − 1.02·47-s − 1.85·49-s − 2.19·53-s + 0.809·55-s − 2.21·59-s + 2.56·61-s − 2.44·67-s + 0.474·71-s + 1.40·73-s − 0.341·77-s + 0.562·79-s − 0.878·83-s + 1.08·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 16 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_h_q |
| 7 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 17 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ab_o_ar |
| 13 | $S_4\times C_2$ | \( 1 + 7 T^{2} + 44 T^{3} + 7 p T^{4} + p^{3} T^{6} \) | 3.13.a_h_bs |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 46 T^{2} + 143 T^{3} + 46 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.f_bu_fn |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 112 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_n_ei |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 76 T^{2} + 311 T^{3} + 76 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.h_cy_lz |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 96 T^{2} + 407 T^{3} + 96 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.h_ds_pr |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 89 T^{2} + 236 T^{3} + 89 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.e_dl_jc |
| 37 | $S_4\times C_2$ | \( 1 - 3 T + 82 T^{2} - 235 T^{3} + 82 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ad_de_ajb |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 112 T^{2} + 741 T^{3} + 112 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.j_ei_bcn |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 52 T^{2} + 231 T^{3} + 52 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_ca_ix |
| 47 | $S_4\times C_2$ | \( 1 + 7 T + 104 T^{2} + 471 T^{3} + 104 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.h_ea_sd |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 215 T^{2} + 1648 T^{3} + 215 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.q_ih_clk |
| 59 | $S_4\times C_2$ | \( 1 + 17 T + 140 T^{2} + 869 T^{3} + 140 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.r_fk_bhl |
| 61 | $S_4\times C_2$ | \( 1 - 20 T + 307 T^{2} - 2676 T^{3} + 307 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.au_lv_adyy |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 325 T^{2} + 2916 T^{3} + 325 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.u_mn_eie |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} + 632 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ae_n_yi |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 115 T^{2} - 488 T^{3} + 115 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_el_asu |
| 79 | $S_4\times C_2$ | \( 1 - 5 T + 144 T^{2} - 855 T^{3} + 144 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.af_fo_abgx |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 844 T^{3} + 17 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.i_r_abgm |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 119 T^{2} + 996 T^{3} + 119 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.o_ep_bmi |
| 97 | $S_4\times C_2$ | \( 1 - 25 T + 490 T^{2} - 5349 T^{3} + 490 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.az_sw_ahxt |
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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
−8.388817235796753340843852673506, −7.947947772723971557850626990425, −7.919199248314929662488144270427, −7.71621992490378090186390731890, −7.35937219981971213627784676536, −7.15702501391858418156981832842, −6.80932895619649158754049101601, −6.39128624324546302019439973267, −6.25431025143851346244979821804, −6.20489907401424747999600509942, −5.69425936125755713942455745990, −5.34766471034992006784600065196, −5.10569778516662827816805260753, −4.83616737176584046610383890445, −4.54770639860561807333143712923, −4.43652110165151585712784055853, −3.77542488029579990015272058254, −3.70036075100916682388973480734, −3.59433377876379816263579762206, −3.08850679422730067224727809645, −2.51112410087327548657469084032, −2.50615285727192729522211204564, −1.78906920493724777266538722074, −1.72267054243700018213379493248, −1.35155626779663788441515821688, 0, 0, 0,
1.35155626779663788441515821688, 1.72267054243700018213379493248, 1.78906920493724777266538722074, 2.50615285727192729522211204564, 2.51112410087327548657469084032, 3.08850679422730067224727809645, 3.59433377876379816263579762206, 3.70036075100916682388973480734, 3.77542488029579990015272058254, 4.43652110165151585712784055853, 4.54770639860561807333143712923, 4.83616737176584046610383890445, 5.10569778516662827816805260753, 5.34766471034992006784600065196, 5.69425936125755713942455745990, 6.20489907401424747999600509942, 6.25431025143851346244979821804, 6.39128624324546302019439973267, 6.80932895619649158754049101601, 7.15702501391858418156981832842, 7.35937219981971213627784676536, 7.71621992490378090186390731890, 7.919199248314929662488144270427, 7.947947772723971557850626990425, 8.388817235796753340843852673506
