| L(s) = 1 | + 6·3-s − 5·4-s − 6·7-s + 14·9-s − 6·11-s − 30·12-s + 6·13-s + 10·16-s − 2·17-s − 2·19-s − 36·21-s + 8·23-s + 14·27-s + 30·28-s − 8·29-s + 4·31-s + 2·32-s − 36·33-s − 70·36-s + 22·37-s + 36·39-s + 4·41-s + 30·43-s + 30·44-s + 16·47-s + 60·48-s + 21·49-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 5/2·4-s − 2.26·7-s + 14/3·9-s − 1.80·11-s − 8.66·12-s + 1.66·13-s + 5/2·16-s − 0.485·17-s − 0.458·19-s − 7.85·21-s + 1.66·23-s + 2.69·27-s + 5.66·28-s − 1.48·29-s + 0.718·31-s + 0.353·32-s − 6.26·33-s − 11.6·36-s + 3.61·37-s + 5.76·39-s + 0.624·41-s + 4.57·43-s + 4.52·44-s + 2.33·47-s + 8.66·48-s + 3·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.18737318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.18737318\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 7 | \( ( 1 + T )^{6} \) | |
| 11 | \( ( 1 + T )^{6} \) | |
| good | 2 | \( 1 + 5 T^{2} + 15 T^{4} - p T^{5} + 35 T^{6} - p^{2} T^{7} + 15 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) | 6.2.a_f_a_p_ac_bj |
| 3 | \( 1 - 2 p T + 22 T^{2} - 62 T^{3} + 49 p T^{4} - 302 T^{5} + 554 T^{6} - 302 p T^{7} + 49 p^{3} T^{8} - 62 p^{3} T^{9} + 22 p^{4} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \) | 6.3.ag_w_ack_fr_alq_vi |
| 13 | \( 1 - 6 T + 56 T^{2} - 288 T^{3} + 1447 T^{4} - 508 p T^{5} + 1802 p T^{6} - 508 p^{2} T^{7} + 1447 p^{2} T^{8} - 288 p^{3} T^{9} + 56 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.ag_ce_alc_cdr_ajua_bira |
| 17 | \( 1 + 2 T + 62 T^{2} + 22 T^{3} + 1571 T^{4} - 1630 T^{5} + 27366 T^{6} - 1630 p T^{7} + 1571 p^{2} T^{8} + 22 p^{3} T^{9} + 62 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.c_ck_w_cil_acks_bomo |
| 19 | \( 1 + 2 T + 4 p T^{2} + 86 T^{3} + 2531 T^{4} + 1452 T^{5} + 55096 T^{6} + 1452 p T^{7} + 2531 p^{2} T^{8} + 86 p^{3} T^{9} + 4 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.19.c_cy_di_dtj_cdw_ddnc |
| 23 | \( 1 - 8 T + 110 T^{2} - 760 T^{3} + 5735 T^{4} - 31432 T^{5} + 171776 T^{6} - 31432 p T^{7} + 5735 p^{2} T^{8} - 760 p^{3} T^{9} + 110 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.ai_eg_abdg_imp_abumy_jucu |
| 29 | \( 1 + 8 T + 126 T^{2} + 816 T^{3} + 7767 T^{4} + 40296 T^{5} + 287348 T^{6} + 40296 p T^{7} + 7767 p^{2} T^{8} + 816 p^{3} T^{9} + 126 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.i_ew_bfk_lmt_chpw_qjbw |
| 31 | \( 1 - 4 T + 118 T^{2} - 374 T^{3} + 6895 T^{4} - 17804 T^{5} + 261070 T^{6} - 17804 p T^{7} + 6895 p^{2} T^{8} - 374 p^{3} T^{9} + 118 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.ae_eo_aok_kff_abaiu_owfe |
| 37 | \( 1 - 22 T + 290 T^{2} - 2614 T^{3} + 19503 T^{4} - 126044 T^{5} + 788552 T^{6} - 126044 p T^{7} + 19503 p^{2} T^{8} - 2614 p^{3} T^{9} + 290 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.aw_le_adwo_bcwd_ahelw_bswmy |
| 41 | \( 1 - 4 T + 142 T^{2} + 26 T^{3} + 6387 T^{4} + 36896 T^{5} + 184210 T^{6} + 36896 p T^{7} + 6387 p^{2} T^{8} + 26 p^{3} T^{9} + 142 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.ae_fm_ba_jlr_ccpc_kmna |
| 43 | \( 1 - 30 T + 550 T^{2} - 7094 T^{3} + 73047 T^{4} - 614216 T^{5} + 4386760 T^{6} - 614216 p T^{7} + 73047 p^{2} T^{8} - 7094 p^{3} T^{9} + 550 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.abe_ve_akmw_eebn_abiyps_jppho |
| 47 | \( 1 - 16 T + 4 p T^{2} - 1390 T^{3} + 7587 T^{4} - 39300 T^{5} + 184014 T^{6} - 39300 p T^{7} + 7587 p^{2} T^{8} - 1390 p^{3} T^{9} + 4 p^{5} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.aq_hg_acbm_lfv_acgdo_kmfm |
| 53 | \( 1 - 6 T + 242 T^{2} - 838 T^{3} + 23975 T^{4} - 47612 T^{5} + 1480424 T^{6} - 47612 p T^{7} + 23975 p^{2} T^{8} - 838 p^{3} T^{9} + 242 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.ag_ji_abgg_bjmd_acslg_dgfzk |
| 59 | \( 1 - 14 T + 258 T^{2} - 2616 T^{3} + 31991 T^{4} - 253492 T^{5} + 2319686 T^{6} - 253492 p T^{7} + 31991 p^{2} T^{8} - 2616 p^{3} T^{9} + 258 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.ao_jy_adwq_bvil_aokzs_fbzms |
| 61 | \( 1 - 12 T + 4 p T^{2} - 2968 T^{3} + 31131 T^{4} - 313642 T^{5} + 2438062 T^{6} - 313642 p T^{7} + 31131 p^{2} T^{8} - 2968 p^{3} T^{9} + 4 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.am_jk_aeke_bubj_arvze_fispq |
| 67 | \( 1 - 46 T + 1190 T^{2} - 21474 T^{3} + 297119 T^{4} - 3283948 T^{5} + 29665008 T^{6} - 3283948 p T^{7} + 297119 p^{2} T^{8} - 21474 p^{3} T^{9} + 1190 p^{4} T^{10} - 46 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.abu_btu_abfty_qxnr_ahevxs_cmxvdw |
| 71 | \( 1 - 4 T + 294 T^{2} - 980 T^{3} + 601 p T^{4} - 119680 T^{5} + 3799268 T^{6} - 119680 p T^{7} + 601 p^{3} T^{8} - 980 p^{3} T^{9} + 294 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.ae_li_abls_cldf_agvbc_iiefs |
| 73 | \( 1 + 4 T + 14 T^{2} - 80 T^{3} + 3935 T^{4} - 37794 T^{5} - 375442 T^{6} - 37794 p T^{7} + 3935 p^{2} T^{8} - 80 p^{3} T^{9} + 14 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.e_o_adc_fvj_acdxq_avjkc |
| 79 | \( 1 + 8 T + 214 T^{2} + 4 p T^{3} + 20183 T^{4} + 13040 T^{5} + 2056160 T^{6} + 13040 p T^{7} + 20183 p^{2} T^{8} + 4 p^{4} T^{9} + 214 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.i_ig_me_bdwh_tho_emzrc |
| 83 | \( 1 - 14 T + 468 T^{2} - 4714 T^{3} + 89003 T^{4} - 687092 T^{5} + 9471224 T^{6} - 687092 p T^{7} + 89003 p^{2} T^{8} - 4714 p^{3} T^{9} + 468 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.ao_sa_agzi_fbrf_abnckq_uswrw |
| 89 | \( 1 - 8 T + 180 T^{2} - 1480 T^{3} + 27891 T^{4} - 189592 T^{5} + 2425768 T^{6} - 189592 p T^{7} + 27891 p^{2} T^{8} - 1480 p^{3} T^{9} + 180 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.ai_gy_acey_bpgt_akuma_fiaku |
| 97 | \( 1 - 42 T + 1212 T^{2} - 23906 T^{3} + 383075 T^{4} - 4888692 T^{5} + 53099384 T^{6} - 4888692 p T^{7} + 383075 p^{2} T^{8} - 23906 p^{3} T^{9} + 1212 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.abq_buq_abjjm_vurr_aksduq_emfdka |
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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
−4.75343307611408058159537022852, −4.38641905605778099809515008755, −4.37407803625745751778328936806, −4.14654769866611478432080654359, −4.06922143052890820270759777976, −3.99339431785435899514737180065, −3.99235145374420795072922653856, −3.61964417242566639048810212929, −3.44380172284402013902192113269, −3.38979068515126659422315839658, −3.34455918323432408276127780389, −3.09475908147918248961427228949, −2.82577066474084596927806257791, −2.66497211494344863383622904270, −2.47455102973592910919723627092, −2.44360268667918342811516790917, −2.42107600721470195520287973795, −2.36131384852626216120314540154, −1.94156794369269749446054082607, −1.84506110063473773310184765304, −0.946481181364501097301341466215, −0.874107618844005677749972845020, −0.73095829784914148669521632323, −0.62021219543234726263114444114, −0.57921547763530838294032337675,
0.57921547763530838294032337675, 0.62021219543234726263114444114, 0.73095829784914148669521632323, 0.874107618844005677749972845020, 0.946481181364501097301341466215, 1.84506110063473773310184765304, 1.94156794369269749446054082607, 2.36131384852626216120314540154, 2.42107600721470195520287973795, 2.44360268667918342811516790917, 2.47455102973592910919723627092, 2.66497211494344863383622904270, 2.82577066474084596927806257791, 3.09475908147918248961427228949, 3.34455918323432408276127780389, 3.38979068515126659422315839658, 3.44380172284402013902192113269, 3.61964417242566639048810212929, 3.99235145374420795072922653856, 3.99339431785435899514737180065, 4.06922143052890820270759777976, 4.14654769866611478432080654359, 4.37407803625745751778328936806, 4.38641905605778099809515008755, 4.75343307611408058159537022852
Plot not available for L-functions of degree greater than 10.
