| L(s) = 1 | + 3-s − 3·5-s − 3·7-s − 3·9-s + 6·11-s − 13-s − 3·15-s + 5·17-s + 2·19-s − 3·21-s + 11·23-s + 6·25-s + 27-s − 3·29-s + 31-s + 6·33-s + 9·35-s + 14·37-s − 39-s + 19·43-s + 9·45-s + 10·47-s − 11·49-s + 5·51-s + 9·53-s − 18·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.13·7-s − 9-s + 1.80·11-s − 0.277·13-s − 0.774·15-s + 1.21·17-s + 0.458·19-s − 0.654·21-s + 2.29·23-s + 6/5·25-s + 0.192·27-s − 0.557·29-s + 0.179·31-s + 1.04·33-s + 1.52·35-s + 2.30·37-s − 0.160·39-s + 2.89·43-s + 1.34·45-s + 1.45·47-s − 1.57·49-s + 0.700·51-s + 1.23·53-s − 2.42·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.088218928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.088218928\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 4 T^{2} - 8 T^{3} + 4 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_e_ai |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 20 T^{2} + 38 T^{3} + 20 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_u_bm |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 29 T^{2} - 100 T^{3} + 29 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bd_adw |
| 13 | $S_4\times C_2$ | \( 1 + T + 34 T^{2} + 28 T^{3} + 34 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.13.b_bi_bc |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 28 T^{2} - 114 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.af_bc_aek |
| 19 | $S_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 148 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ac_ad_fs |
| 23 | $S_4\times C_2$ | \( 1 - 11 T + 94 T^{2} - 522 T^{3} + 94 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.al_dq_auc |
| 31 | $S_4\times C_2$ | \( 1 - T + 74 T^{2} - 30 T^{3} + 74 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ab_cw_abe |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 155 T^{2} - 1068 T^{3} + 155 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ao_fz_abpc |
| 41 | $S_4\times C_2$ | \( 1 + 107 T^{2} + 8 T^{3} + 107 p T^{4} + p^{3} T^{6} \) | 3.41.a_ed_i |
| 43 | $S_4\times C_2$ | \( 1 - 19 T + 230 T^{2} - 1740 T^{3} + 230 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.at_iw_acoy |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 149 T^{2} - 908 T^{3} + 149 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ak_ft_abiy |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 150 T^{2} - 900 T^{3} + 150 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_fu_abiq |
| 59 | $S_4\times C_2$ | \( 1 - T + 40 T^{2} + 6 T^{3} + 40 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ab_bo_g |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 68 T^{2} - 420 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ah_cq_aqe |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 121 T^{2} - 944 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ai_er_abki |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 197 T^{2} - 1760 T^{3} + 197 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aq_hp_acps |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 68 T^{2} + 82 T^{3} + 68 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_cq_de |
| 79 | $S_4\times C_2$ | \( 1 - 7 T + 196 T^{2} - 898 T^{3} + 196 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ah_ho_abio |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 261 T^{2} - 1596 T^{3} + 261 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ak_kb_acjk |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 379 T^{2} + 3744 T^{3} + 379 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_op_foa |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 236 T^{2} + 1634 T^{3} + 236 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.j_jc_ckw |
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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
−8.873137139267414334363871356604, −8.353179108099813709024994869793, −8.148168902366915092959029037878, −7.943422588194208558736525177094, −7.66778404285082987604523138767, −7.27364383131908518408581038725, −7.05832376917290536549913584991, −6.79317146596486714981716575068, −6.44799923724852378973767569610, −6.39375030488883210909882357222, −5.79321255513296329438880280551, −5.46794066116096750757168248611, −5.42613398237589521539004847122, −4.85483874756190950598511535309, −4.43464376771744946532242004000, −4.22536245560309521049637709095, −3.68051149256649893344313567840, −3.64113027009450492130405642219, −3.38193587655715672682786422953, −2.78538836922409242930771405234, −2.62817130604426132762915107719, −2.45837221119487774530932993239, −1.33872183359633887123592215886, −0.808797827599327022244923399345, −0.76731740118093849866248853100,
0.76731740118093849866248853100, 0.808797827599327022244923399345, 1.33872183359633887123592215886, 2.45837221119487774530932993239, 2.62817130604426132762915107719, 2.78538836922409242930771405234, 3.38193587655715672682786422953, 3.64113027009450492130405642219, 3.68051149256649893344313567840, 4.22536245560309521049637709095, 4.43464376771744946532242004000, 4.85483874756190950598511535309, 5.42613398237589521539004847122, 5.46794066116096750757168248611, 5.79321255513296329438880280551, 6.39375030488883210909882357222, 6.44799923724852378973767569610, 6.79317146596486714981716575068, 7.05832376917290536549913584991, 7.27364383131908518408581038725, 7.66778404285082987604523138767, 7.943422588194208558736525177094, 8.148168902366915092959029037878, 8.353179108099813709024994869793, 8.873137139267414334363871356604
