| L(s) = 1 | − 2-s − 4·4-s + 5·8-s + 16·11-s + 3·13-s + 4·16-s + 2·17-s − 11·19-s − 16·22-s − 7·23-s − 3·26-s + 15·29-s − 13·31-s − 9·32-s − 2·34-s − 5·37-s + 11·38-s + 13·41-s + 7·43-s − 64·44-s + 7·46-s − 2·47-s − 16·49-s − 12·52-s + 20·53-s − 15·58-s + 32·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2·4-s + 1.76·8-s + 4.82·11-s + 0.832·13-s + 16-s + 0.485·17-s − 2.52·19-s − 3.41·22-s − 1.45·23-s − 0.588·26-s + 2.78·29-s − 2.33·31-s − 1.59·32-s − 0.342·34-s − 0.821·37-s + 1.78·38-s + 2.03·41-s + 1.06·43-s − 9.64·44-s + 1.03·46-s − 0.291·47-s − 2.28·49-s − 1.66·52-s + 2.74·53-s − 1.96·58-s + 4.16·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.831312509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.831312509\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 37 | $C_1$ | \( ( 1 + T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + T + 5 T^{2} + p^{2} T^{3} + 15 T^{4} + 11 T^{5} + 15 p T^{6} + p^{4} T^{7} + 5 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.2.b_f_e_p_l |
| 7 | $C_2 \wr S_5$ | \( 1 + 16 T^{2} - 12 T^{3} + 183 T^{4} - 57 T^{5} + 183 p T^{6} - 12 p^{2} T^{7} + 16 p^{3} T^{8} + p^{5} T^{10} \) | 5.7.a_q_am_hb_acf |
| 11 | $C_2 \wr S_5$ | \( 1 - 16 T + 138 T^{2} - 823 T^{3} + 3776 T^{4} - 13879 T^{5} + 3776 p T^{6} - 823 p^{2} T^{7} + 138 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.aq_fi_abfr_fpg_aunv |
| 13 | $C_2 \wr S_5$ | \( 1 - 3 T + 20 T^{2} - 31 T^{3} + 133 T^{4} - 171 T^{5} + 133 p T^{6} - 31 p^{2} T^{7} + 20 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.ad_u_abf_fd_agp |
| 17 | $C_2 \wr S_5$ | \( 1 - 2 T + 21 T^{2} - 89 T^{3} + 690 T^{4} - 1235 T^{5} + 690 p T^{6} - 89 p^{2} T^{7} + 21 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.ac_v_adl_bao_abvn |
| 19 | $C_2 \wr S_5$ | \( 1 + 11 T + 90 T^{2} + 525 T^{3} + 2547 T^{4} + 11653 T^{5} + 2547 p T^{6} + 525 p^{2} T^{7} + 90 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.l_dm_uf_dtz_rgf |
| 23 | $C_2 \wr S_5$ | \( 1 + 7 T + 44 T^{2} + 420 T^{3} + 2338 T^{4} + 9411 T^{5} + 2338 p T^{6} + 420 p^{2} T^{7} + 44 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.h_bs_qe_dly_nxz |
| 29 | $C_2 \wr S_5$ | \( 1 - 15 T + 154 T^{2} - 1103 T^{3} + 6827 T^{4} - 36547 T^{5} + 6827 p T^{6} - 1103 p^{2} T^{7} + 154 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.ap_fy_abql_kcp_accbr |
| 31 | $C_2 \wr S_5$ | \( 1 + 13 T + 116 T^{2} + 910 T^{3} + 5794 T^{4} + 31515 T^{5} + 5794 p T^{6} + 910 p^{2} T^{7} + 116 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.n_em_bja_iow_buqd |
| 41 | $C_2 \wr S_5$ | \( 1 - 13 T + 219 T^{2} - 1885 T^{3} + 17663 T^{4} - 110047 T^{5} + 17663 p T^{6} - 1885 p^{2} T^{7} + 219 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.an_il_acun_badj_aggup |
| 43 | $C_2 \wr S_5$ | \( 1 - 7 T + 176 T^{2} - 978 T^{3} + 13616 T^{4} - 58423 T^{5} + 13616 p T^{6} - 978 p^{2} T^{7} + 176 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.ah_gu_ablq_uds_adilb |
| 47 | $C_2 \wr S_5$ | \( 1 + 2 T + 122 T^{2} + 338 T^{3} + 9201 T^{4} + 17179 T^{5} + 9201 p T^{6} + 338 p^{2} T^{7} + 122 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.c_es_na_npx_zkt |
| 53 | $C_2 \wr S_5$ | \( 1 - 20 T + 250 T^{2} - 2405 T^{3} + 18710 T^{4} - 135769 T^{5} + 18710 p T^{6} - 2405 p^{2} T^{7} + 250 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.au_jq_adon_bbrq_ahsvx |
| 59 | $C_2 \wr S_5$ | \( 1 - 32 T + 630 T^{2} - 8624 T^{3} + 93123 T^{4} - 792761 T^{5} + 93123 p T^{6} - 8624 p^{2} T^{7} + 630 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.abg_yg_amts_fhtr_abtcsv |
| 61 | $C_2 \wr S_5$ | \( 1 - T + 252 T^{2} - 106 T^{3} + 27798 T^{4} - 6223 T^{5} + 27798 p T^{6} - 106 p^{2} T^{7} + 252 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.61.ab_js_aec_bpde_ajfj |
| 67 | $C_2 \wr S_5$ | \( 1 - 3 T + 285 T^{2} - 663 T^{3} + 35341 T^{4} - 63215 T^{5} + 35341 p T^{6} - 663 p^{2} T^{7} + 285 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.ad_kz_azn_cahh_adpnj |
| 71 | $C_2 \wr S_5$ | \( 1 - 26 T + 572 T^{2} - 7918 T^{3} + 95961 T^{4} - 859577 T^{5} + 95961 p T^{6} - 7918 p^{2} T^{7} + 572 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.aba_wa_also_flyv_abwxor |
| 73 | $C_2 \wr S_5$ | \( 1 + 12 T + 173 T^{2} + 248 T^{3} - 2070 T^{4} - 107912 T^{5} - 2070 p T^{6} + 248 p^{2} T^{7} + 173 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.m_gr_jo_adbq_agdqm |
| 79 | $C_2 \wr S_5$ | \( 1 + 3 T + 125 T^{2} - 63 T^{3} + 14557 T^{4} + 29471 T^{5} + 14557 p T^{6} - 63 p^{2} T^{7} + 125 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.d_ev_acl_vnx_brpn |
| 83 | $C_2 \wr S_5$ | \( 1 - 14 T + 347 T^{2} - 3869 T^{3} + 54402 T^{4} - 453027 T^{5} + 54402 p T^{6} - 3869 p^{2} T^{7} + 347 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.ao_nj_afsv_dcmk_azued |
| 89 | $C_2 \wr S_5$ | \( 1 - 20 T + 413 T^{2} - 5199 T^{3} + 68868 T^{4} - 644139 T^{5} + 68868 p T^{6} - 5199 p^{2} T^{7} + 413 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.au_px_ahrz_dxwu_abkqwp |
| 97 | $C_2 \wr S_5$ | \( 1 - 15 T + 218 T^{2} - 446 T^{3} + 2600 T^{4} + 110943 T^{5} + 2600 p T^{6} - 446 p^{2} T^{7} + 218 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.ap_ik_are_dwa_gidb |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.36475267463626035089634346725, −4.26871706780659804782093552494, −4.17414005193788208540729496431, −4.15221668383087956334649951724, −4.01105588690945750865985198089, −3.80595571746826355500786610401, −3.79246701598885184233085181716, −3.63989309322251655798722940741, −3.56339341431338388486832733897, −3.44239044163863484018948152748, −2.95070744048078442920652404619, −2.86795046494324904497889465759, −2.73558937701165853347356407623, −2.26634014894010499707920485522, −2.08491715199581617886913290394, −2.06712875927414212374639836244, −1.96576534992836944839991058189, −1.80726268417622310860856806404, −1.45081650970951734405380678143, −1.25814097475873766987355162032, −0.906800132124923830822231339161, −0.841177096520293641013278362841, −0.836851591011418800736857589702, −0.49157498891012072647486336674, −0.30194435918006541160022081712,
0.30194435918006541160022081712, 0.49157498891012072647486336674, 0.836851591011418800736857589702, 0.841177096520293641013278362841, 0.906800132124923830822231339161, 1.25814097475873766987355162032, 1.45081650970951734405380678143, 1.80726268417622310860856806404, 1.96576534992836944839991058189, 2.06712875927414212374639836244, 2.08491715199581617886913290394, 2.26634014894010499707920485522, 2.73558937701165853347356407623, 2.86795046494324904497889465759, 2.95070744048078442920652404619, 3.44239044163863484018948152748, 3.56339341431338388486832733897, 3.63989309322251655798722940741, 3.79246701598885184233085181716, 3.80595571746826355500786610401, 4.01105588690945750865985198089, 4.15221668383087956334649951724, 4.17414005193788208540729496431, 4.26871706780659804782093552494, 4.36475267463626035089634346725
