| L(s) = 1 | + 6·3-s − 4·4-s + 6·5-s − 2·7-s + 21·9-s − 12·11-s − 24·12-s − 4·13-s + 36·15-s + 6·16-s − 14·17-s − 8·19-s − 24·20-s − 12·21-s + 21·25-s + 56·27-s + 8·28-s − 14·29-s − 6·31-s − 4·32-s − 72·33-s − 12·35-s − 84·36-s + 2·37-s − 24·39-s − 2·41-s − 12·43-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 2·4-s + 2.68·5-s − 0.755·7-s + 7·9-s − 3.61·11-s − 6.92·12-s − 1.10·13-s + 9.29·15-s + 3/2·16-s − 3.39·17-s − 1.83·19-s − 5.36·20-s − 2.61·21-s + 21/5·25-s + 10.7·27-s + 1.51·28-s − 2.59·29-s − 1.07·31-s − 0.707·32-s − 12.5·33-s − 2.02·35-s − 14·36-s + 0.328·37-s − 3.84·39-s − 0.312·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( ( 1 - T )^{6} \) | |
| 5 | \( ( 1 - T )^{6} \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T^{2} + 5 p T^{4} + p^{2} T^{5} + 11 p T^{6} + p^{3} T^{7} + 5 p^{3} T^{8} + p^{6} T^{10} + p^{6} T^{12} \) | 6.2.a_e_a_k_e_w |
| 7 | \( 1 + 2 T + 23 T^{2} + 46 T^{3} + 38 p T^{4} + 530 T^{5} + 2115 T^{6} + 530 p T^{7} + 38 p^{3} T^{8} + 46 p^{3} T^{9} + 23 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.c_x_bu_kg_uk_ddj |
| 11 | \( 1 + 12 T + 100 T^{2} + 636 T^{3} + 3233 T^{4} + 13700 T^{5} + 49590 T^{6} + 13700 p T^{7} + 3233 p^{2} T^{8} + 636 p^{3} T^{9} + 100 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.m_dw_ym_euj_ugy_cvji |
| 13 | \( 1 + 4 T + 48 T^{2} + 196 T^{3} + 1307 T^{4} + 4272 T^{5} + 21590 T^{6} + 4272 p T^{7} + 1307 p^{2} T^{8} + 196 p^{3} T^{9} + 48 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.e_bw_ho_byh_gii_bfyk |
| 17 | \( 1 + 14 T + 151 T^{2} + 1138 T^{3} + 7300 T^{4} + 37790 T^{5} + 171395 T^{6} + 37790 p T^{7} + 7300 p^{2} T^{8} + 1138 p^{3} T^{9} + 151 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.o_fv_bru_kuu_cdxm_jtod |
| 19 | \( 1 + 8 T + 6 p T^{2} + 36 p T^{3} + 5311 T^{4} + 24588 T^{5} + 133310 T^{6} + 24588 p T^{7} + 5311 p^{2} T^{8} + 36 p^{4} T^{9} + 6 p^{5} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.19.i_ek_bai_hwh_bkjs_hpfi |
| 29 | \( 1 + 14 T + 127 T^{2} + 34 p T^{3} + 6630 T^{4} + 40086 T^{5} + 229161 T^{6} + 40086 p T^{7} + 6630 p^{2} T^{8} + 34 p^{4} T^{9} + 127 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.o_ex_bly_jva_chhu_nazx |
| 31 | \( 1 + 6 T + 121 T^{2} + 426 T^{3} + 5382 T^{4} + 10406 T^{5} + 160897 T^{6} + 10406 p T^{7} + 5382 p^{2} T^{8} + 426 p^{3} T^{9} + 121 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.g_er_qk_hza_pkg_jeaj |
| 37 | \( 1 - 2 T + 103 T^{2} - 126 T^{3} + 6306 T^{4} - 8114 T^{5} + 287759 T^{6} - 8114 p T^{7} + 6306 p^{2} T^{8} - 126 p^{3} T^{9} + 103 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.ac_dz_aew_jio_amac_qjrr |
| 41 | \( 1 + 2 T + 81 T^{2} - 158 T^{3} + 2152 T^{4} - 27870 T^{5} + 8697 T^{6} - 27870 p T^{7} + 2152 p^{2} T^{8} - 158 p^{3} T^{9} + 81 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.c_dd_agc_deu_abpfy_mwn |
| 43 | \( 1 + 12 T + 242 T^{2} + 2280 T^{3} + 25051 T^{4} + 184676 T^{5} + 1417552 T^{6} + 184676 p T^{7} + 25051 p^{2} T^{8} + 2280 p^{3} T^{9} + 242 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.m_ji_djs_blbn_kney_dcqzg |
| 47 | \( 1 - 8 T + 152 T^{2} - 1052 T^{3} + 11881 T^{4} - 73536 T^{5} + 663642 T^{6} - 73536 p T^{7} + 11881 p^{2} T^{8} - 1052 p^{3} T^{9} + 152 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.ai_fw_abom_roz_aeeui_bltss |
| 53 | \( 1 + 14 T + 317 T^{2} + 3258 T^{3} + 42066 T^{4} + 325318 T^{5} + 2970611 T^{6} + 325318 p T^{7} + 42066 p^{2} T^{8} + 3258 p^{3} T^{9} + 317 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.o_mf_evi_ckfy_sngg_gnakh |
| 59 | \( 1 + 14 T + 363 T^{2} + 3550 T^{3} + 51742 T^{4} + 381846 T^{5} + 3990069 T^{6} + 381846 p T^{7} + 51742 p^{2} T^{8} + 3550 p^{3} T^{9} + 363 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.o_nz_fgo_cyoc_vswk_itamf |
| 61 | \( 1 - 8 T + 200 T^{2} - 1308 T^{3} + 19207 T^{4} - 92852 T^{5} + 1303522 T^{6} - 92852 p T^{7} + 19207 p^{2} T^{8} - 1308 p^{3} T^{9} + 200 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.ai_hs_abyi_bckt_afhjg_cwehm |
| 67 | \( 1 - 2 T + 59 T^{2} + 406 T^{3} + 4506 T^{4} + 35734 T^{5} + 338939 T^{6} + 35734 p T^{7} + 4506 p^{2} T^{8} + 406 p^{3} T^{9} + 59 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.ac_ch_pq_gri_cawk_thkd |
| 71 | \( 1 + 10 T + 237 T^{2} + 1746 T^{3} + 29868 T^{4} + 180386 T^{5} + 2453733 T^{6} + 180386 p T^{7} + 29868 p^{2} T^{8} + 1746 p^{3} T^{9} + 237 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.k_jd_cpe_bseu_kgvy_fjpuj |
| 73 | \( 1 + 4 T + 196 T^{2} - 268 T^{3} + 9103 T^{4} - 159096 T^{5} + 23302 T^{6} - 159096 p T^{7} + 9103 p^{2} T^{8} - 268 p^{3} T^{9} + 196 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.e_ho_aki_nmd_ajbjc_bimg |
| 79 | \( 1 - 16 T + 394 T^{2} - 4560 T^{3} + 67263 T^{4} - 622720 T^{5} + 6778892 T^{6} - 622720 p T^{7} + 67263 p^{2} T^{8} - 4560 p^{3} T^{9} + 394 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.aq_pe_agtk_dvnb_abjleu_ovryq |
| 83 | \( 1 - 10 T + 395 T^{2} - 3942 T^{3} + 71552 T^{4} - 633682 T^{5} + 7578303 T^{6} - 633682 p T^{7} + 71552 p^{2} T^{8} - 3942 p^{3} T^{9} + 395 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.ak_pf_afvq_ebwa_abkbkk_qpenf |
| 89 | \( 1 + 16 T + 416 T^{2} + 5528 T^{3} + 79603 T^{4} + 855528 T^{5} + 8997264 T^{6} + 855528 p T^{7} + 79603 p^{2} T^{8} + 5528 p^{3} T^{9} + 416 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.q_qa_ieq_entr_bwroy_trxoq |
| 97 | \( 1 + 8 T + 278 T^{2} + 2028 T^{3} + 47327 T^{4} + 3148 p T^{5} + 5389392 T^{6} + 3148 p^{2} T^{7} + 47327 p^{2} T^{8} + 2028 p^{3} T^{9} + 278 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.i_ks_daa_csah_rjsm_luqmi |
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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
−4.46791709118167594868290868508, −4.26619268039987353628758370925, −4.05557986008278231273439498150, −3.95901680618484646956023379038, −3.74765492645953129586195606215, −3.67744846987194292164862133619, −3.59925559949222839292227270829, −3.35813586654292297084568227614, −3.18661695878075570850655287260, −3.12824367012157628260917790289, −3.07714169440888167348196589655, −2.87242926158464328987761475401, −2.58788946138009423763632618378, −2.48432064751675757561379201132, −2.43241776605630186931379920211, −2.28394556859152745512399576843, −2.26605263942012270640460399019, −2.16068426431266249206491514361, −2.10946210830268008028636265187, −1.89391975192769330057344345534, −1.70779569345026947760004560076, −1.36598898236577479239983657297, −1.35573493173935979469888687490, −1.26047203888461618099097686437, −1.03801743643561482568392540611, 0, 0, 0, 0, 0, 0,
1.03801743643561482568392540611, 1.26047203888461618099097686437, 1.35573493173935979469888687490, 1.36598898236577479239983657297, 1.70779569345026947760004560076, 1.89391975192769330057344345534, 2.10946210830268008028636265187, 2.16068426431266249206491514361, 2.26605263942012270640460399019, 2.28394556859152745512399576843, 2.43241776605630186931379920211, 2.48432064751675757561379201132, 2.58788946138009423763632618378, 2.87242926158464328987761475401, 3.07714169440888167348196589655, 3.12824367012157628260917790289, 3.18661695878075570850655287260, 3.35813586654292297084568227614, 3.59925559949222839292227270829, 3.67744846987194292164862133619, 3.74765492645953129586195606215, 3.95901680618484646956023379038, 4.05557986008278231273439498150, 4.26619268039987353628758370925, 4.46791709118167594868290868508
Plot not available for L-functions of degree greater than 10.
