| L(s) = 1 | − 2-s + 2·4-s + 5·5-s + 8-s − 5·10-s − 2·11-s + 3·13-s − 24·17-s + 6·19-s + 10·20-s + 2·22-s + 17·25-s − 3·26-s + 29-s − 3·31-s + 4·32-s + 24·34-s − 6·37-s − 6·38-s + 5·40-s + 22·41-s + 3·43-s − 4·44-s + 9·47-s − 17·50-s + 6·52-s + 36·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s + 2.23·5-s + 0.353·8-s − 1.58·10-s − 0.603·11-s + 0.832·13-s − 5.82·17-s + 1.37·19-s + 2.23·20-s + 0.426·22-s + 17/5·25-s − 0.588·26-s + 0.185·29-s − 0.538·31-s + 0.707·32-s + 4.11·34-s − 0.986·37-s − 0.973·38-s + 0.790·40-s + 3.43·41-s + 0.457·43-s − 0.603·44-s + 1.31·47-s − 2.40·50-s + 0.832·52-s + 4.94·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.718310108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.718310108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + T - T^{2} - p^{2} T^{3} - 3 T^{4} + p T^{5} + 13 T^{6} + p^{2} T^{7} - 3 p^{2} T^{8} - p^{5} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | 6.2.b_ab_ae_ad_c_n |
| 5 | \( 1 - p T + 8 T^{2} - 7 T^{3} + 9 T^{4} + 62 T^{5} - 299 T^{6} + 62 p T^{7} + 9 p^{2} T^{8} - 7 p^{3} T^{9} + 8 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \) | 6.5.af_i_ah_j_ck_aln |
| 11 | \( 1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 416 p T^{7} + 48 p^{2} T^{8} + 34 p^{3} T^{9} - 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.c_ak_bi_bw_aqa_bf |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) | 6.13.ad_abh_bg_blt_avd_aulm |
| 17 | \( ( 1 + 12 T + 90 T^{2} + 435 T^{3} + 90 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.17.y_mm_emo_bfyu_gykq_bfszj |
| 19 | \( ( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.19.ag_eh_aua_hrh_abbys_hlbj |
| 23 | \( 1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 324 p T^{7} + 468 p^{2} T^{8} + 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \) | 6.23.a_abk_s_sa_amm_ahzl |
| 29 | \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) | 6.29.ab_ade_bf_gof_abde_ailvf |
| 31 | \( 1 + 3 T - 60 T^{2} - 219 T^{3} + 1983 T^{4} + 4746 T^{5} - 51289 T^{6} + 4746 p T^{7} + 1983 p^{2} T^{8} - 219 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.d_aci_ail_cyh_hao_acxwr |
| 37 | \( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.37.g_et_bkm_ntd_ddze_babul |
| 41 | \( 1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 135116 p T^{7} + 18432 p^{2} T^{8} - 1802 p^{3} T^{9} + 206 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.aw_hy_acri_bbgy_ahrwu_btzqh |
| 43 | \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ad_acc_vx_et_atmo_goql |
| 47 | \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.aj_ag_ul_adpp_afcg_fytz |
| 53 | \( ( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.53.abk_bem_asdm_ilui_adeffk_zjvix |
| 59 | \( 1 - 9 T - 90 T^{2} + 459 T^{3} + 10161 T^{4} - 20556 T^{5} - 598421 T^{6} - 20556 p T^{7} + 10161 p^{2} T^{8} + 459 p^{3} T^{9} - 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.aj_adm_rr_pav_abekq_abibgf |
| 61 | \( 1 + 6 T - 126 T^{2} - 358 T^{3} + 12372 T^{4} + 11472 T^{5} - 838653 T^{6} + 11472 p T^{7} + 12372 p^{2} T^{8} - 358 p^{3} T^{9} - 126 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.g_aew_anu_shw_qzg_abvspx |
| 67 | \( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 4098 p T^{7} + 438 p^{2} T^{8} + 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \) | 6.67.a_g_cao_qw_gbq_ciqit |
| 71 | \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.71.s_tb_jbk_fith_bwkqs_tuxil |
| 73 | \( ( 1 + 3 T + 51 T^{2} + 681 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.73.g_eh_cme_uxp_hbco_dtsqh |
| 79 | \( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 51726 p T^{7} - 3225 p^{2} T^{8} - 367 p^{3} T^{9} + 36 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.p_bk_aod_aeub_acynm_abmniv |
| 83 | \( 1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 90444 p T^{7} + 34812 p^{2} T^{8} + 582 p^{3} T^{9} - 144 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.am_afo_wk_bzmy_afduq_afvdqn |
| 89 | \( ( 1 + 2 T + 116 T^{2} + 735 T^{3} + 116 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.89.e_jc_cwk_ccuu_nwfk_jqvtj |
| 97 | \( 1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 78504 p T^{7} + 14223 p^{2} T^{8} - 573 p^{3} T^{9} - 168 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.ad_agm_awb_vbb_emdk_acvvvr |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.20449903360357476104239078869, −4.93638861308477700226044222074, −4.79899852179538048999484675489, −4.62161097739201357511159715691, −4.29560433734837518888377513956, −4.28069310422287707472236740358, −4.04031767573790269977743327848, −3.97533846128081523065531078710, −3.94056782090519340924507797513, −3.90687193712974417022186936887, −3.28671302026344947800047341609, −2.84376931034338381081309340567, −2.83845913256018859175305564517, −2.81182589513844942308985658155, −2.67597004743822502881920085480, −2.49022990805779582779462559220, −2.30016943899736280641930683705, −1.97578825152416051364773576832, −1.83683858060428723570752233424, −1.76617313827421943088596525522, −1.75403876117183729052527973093, −0.961439780392437005350392772403, −0.959147395340411066769004114683, −0.954863224303838284389296525243, −0.19631850582741353447471422301,
0.19631850582741353447471422301, 0.954863224303838284389296525243, 0.959147395340411066769004114683, 0.961439780392437005350392772403, 1.75403876117183729052527973093, 1.76617313827421943088596525522, 1.83683858060428723570752233424, 1.97578825152416051364773576832, 2.30016943899736280641930683705, 2.49022990805779582779462559220, 2.67597004743822502881920085480, 2.81182589513844942308985658155, 2.83845913256018859175305564517, 2.84376931034338381081309340567, 3.28671302026344947800047341609, 3.90687193712974417022186936887, 3.94056782090519340924507797513, 3.97533846128081523065531078710, 4.04031767573790269977743327848, 4.28069310422287707472236740358, 4.29560433734837518888377513956, 4.62161097739201357511159715691, 4.79899852179538048999484675489, 4.93638861308477700226044222074, 5.20449903360357476104239078869
Plot not available for L-functions of degree greater than 10.
