| L(s) = 1 | + 2-s + 6·3-s − 2·4-s + 6·6-s + 2·7-s − 4·8-s + 21·9-s + 7·11-s − 12·12-s + 4·13-s + 2·14-s + 16-s + 21·18-s + 17·19-s + 12·21-s + 7·22-s − 23-s − 24·24-s + 4·26-s + 56·27-s − 4·28-s − 8·29-s − 6·31-s + 32-s + 42·33-s − 42·36-s + 14·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3.46·3-s − 4-s + 2.44·6-s + 0.755·7-s − 1.41·8-s + 7·9-s + 2.11·11-s − 3.46·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 4.94·18-s + 3.90·19-s + 2.61·21-s + 1.49·22-s − 0.208·23-s − 4.89·24-s + 0.784·26-s + 10.7·27-s − 0.755·28-s − 1.48·29-s − 1.07·31-s + 0.176·32-s + 7.31·33-s − 7·36-s + 2.30·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(136.0158622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(136.0158622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( ( 1 + T )^{6} \) | |
| good | 2 | \( 1 - T + 3 T^{2} - T^{3} + p T^{4} + p^{3} T^{5} - 3 p T^{6} + p^{4} T^{7} + p^{3} T^{8} - p^{3} T^{9} + 3 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) | 6.2.ab_d_ab_c_i_ag |
| 7 | \( 1 - 2 T + 2 p T^{2} - 2 p T^{3} + 135 T^{4} - 20 p T^{5} + 1156 T^{6} - 20 p^{2} T^{7} + 135 p^{2} T^{8} - 2 p^{4} T^{9} + 2 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.ac_o_ao_ff_afk_bsm |
| 11 | \( 1 - 7 T + 61 T^{2} - 332 T^{3} + 1643 T^{4} - 6733 T^{5} + 24042 T^{6} - 6733 p T^{7} + 1643 p^{2} T^{8} - 332 p^{3} T^{9} + 61 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.ah_cj_amu_clf_ajyz_bjos |
| 13 | \( 1 - 4 T + 40 T^{2} - 138 T^{3} + 959 T^{4} - 2722 T^{5} + 14176 T^{6} - 2722 p T^{7} + 959 p^{2} T^{8} - 138 p^{3} T^{9} + 40 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.ae_bo_afi_bkx_aeas_uzg |
| 17 | \( 1 + 35 T^{2} - 40 T^{3} + 678 T^{4} - 2280 T^{5} + 10739 T^{6} - 2280 p T^{7} + 678 p^{2} T^{8} - 40 p^{3} T^{9} + 35 p^{4} T^{10} + p^{6} T^{12} \) | 6.17.a_bj_abo_bac_adjs_pxb |
| 19 | \( 1 - 17 T + 185 T^{2} - 1446 T^{3} + 9187 T^{4} - 49093 T^{5} + 228658 T^{6} - 49093 p T^{7} + 9187 p^{2} T^{8} - 1446 p^{3} T^{9} + 185 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \) | 6.19.ar_hd_acdq_npj_acuqf_nago |
| 23 | \( 1 + T + 103 T^{2} + 84 T^{3} + 4941 T^{4} + 3275 T^{5} + 6194 p T^{6} + 3275 p T^{7} + 4941 p^{2} T^{8} + 84 p^{3} T^{9} + 103 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | 6.23.b_dz_dg_hib_evz_icti |
| 29 | \( 1 + 8 T + 109 T^{2} + 520 T^{3} + 5366 T^{4} + 22964 T^{5} + 195945 T^{6} + 22964 p T^{7} + 5366 p^{2} T^{8} + 520 p^{3} T^{9} + 109 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.i_ef_ua_hyk_bhzg_ldwj |
| 37 | \( 1 - 14 T + 240 T^{2} - 2412 T^{3} + 22823 T^{4} - 172326 T^{5} + 1134136 T^{6} - 172326 p T^{7} + 22823 p^{2} T^{8} - 2412 p^{3} T^{9} + 240 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.ao_jg_adou_bhtv_ajuxy_cmnsq |
| 41 | \( 1 - 18 T + 314 T^{2} - 3528 T^{3} + 35963 T^{4} - 282510 T^{5} + 2031164 T^{6} - 282510 p T^{7} + 35963 p^{2} T^{8} - 3528 p^{3} T^{9} + 314 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.as_mc_affs_cbff_aqbxu_elors |
| 43 | \( 1 - 15 T + 241 T^{2} - 2302 T^{3} + 23207 T^{4} - 168867 T^{5} + 1276190 T^{6} - 168867 p T^{7} + 23207 p^{2} T^{8} - 2302 p^{3} T^{9} + 241 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ap_jh_adko_biip_ajpux_cupwg |
| 47 | \( 1 + 13 T + 279 T^{2} + 2512 T^{3} + 30689 T^{4} + 209083 T^{5} + 1859862 T^{6} + 209083 p T^{7} + 30689 p^{2} T^{8} + 2512 p^{3} T^{9} + 279 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.n_kt_dsq_btkj_lxhr_ebvhe |
| 53 | \( 1 + 13 T + 299 T^{2} + 56 p T^{3} + 38639 T^{4} + 294055 T^{5} + 2705310 T^{6} + 294055 p T^{7} + 38639 p^{2} T^{8} + 56 p^{4} T^{9} + 299 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.n_ln_eke_cfed_qszv_fxxyk |
| 59 | \( 1 - 32 T + 746 T^{2} - 11706 T^{3} + 150771 T^{4} - 1520962 T^{5} + 13009868 T^{6} - 1520962 p T^{7} + 150771 p^{2} T^{8} - 11706 p^{3} T^{9} + 746 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.abg_bcs_arig_ipax_adinyo_bcmfjo |
| 61 | \( 1 + 78 T^{2} - 532 T^{3} + 4727 T^{4} - 38412 T^{5} + 489348 T^{6} - 38412 p T^{7} + 4727 p^{2} T^{8} - 532 p^{3} T^{9} + 78 p^{4} T^{10} + p^{6} T^{12} \) | 6.61.a_da_aum_gzv_acevk_bbvxc |
| 67 | \( 1 - 9 T + 99 T^{2} + 554 T^{3} - 639 T^{4} + 36351 T^{5} + 434202 T^{6} + 36351 p T^{7} - 639 p^{2} T^{8} + 554 p^{3} T^{9} + 99 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.aj_dv_vi_ayp_cbud_ysic |
| 71 | \( 1 - T + 217 T^{2} - 214 T^{3} + 24405 T^{4} + 3395 T^{5} + 2012906 T^{6} + 3395 p T^{7} + 24405 p^{2} T^{8} - 214 p^{3} T^{9} + 217 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) | 6.71.ab_ij_aig_bkcr_fap_eknrm |
| 73 | \( 1 - 14 T + 378 T^{2} - 3982 T^{3} + 61583 T^{4} - 510356 T^{5} + 5746956 T^{6} - 510356 p T^{7} + 61583 p^{2} T^{8} - 3982 p^{3} T^{9} + 378 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.ao_oo_afxe_dncp_abdazc_mozku |
| 79 | \( 1 - 13 T + 257 T^{2} - 3018 T^{3} + 41213 T^{4} - 378909 T^{5} + 4022884 T^{6} - 378909 p T^{7} + 41213 p^{2} T^{8} - 3018 p^{3} T^{9} + 257 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.an_jx_aemc_cizd_avonl_iuxai |
| 83 | \( 1 - 9 T + 301 T^{2} - 2372 T^{3} + 35379 T^{4} - 280047 T^{5} + 2870846 T^{6} - 280047 p T^{7} + 35379 p^{2} T^{8} - 2372 p^{3} T^{9} + 301 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.aj_lp_adng_cait_apyhb_ghive |
| 89 | \( 1 - 34 T + 735 T^{2} - 12402 T^{3} + 171034 T^{4} - 1998742 T^{5} + 20216471 T^{6} - 1998742 p T^{7} + 171034 p^{2} T^{8} - 12402 p^{3} T^{9} + 735 p^{4} T^{10} - 34 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.abi_bch_asja_jtag_aejsss_bsggap |
| 97 | \( 1 - 30 T + 909 T^{2} - 16106 T^{3} + 270890 T^{4} - 3275222 T^{5} + 37267781 T^{6} - 3275222 p T^{7} + 270890 p^{2} T^{8} - 16106 p^{3} T^{9} + 909 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.abe_biz_axvm_pksw_ahejac_ddojwf |
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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
−4.65348975009075887704352324318, −4.43069021815222524224207791804, −4.16178965812161801250090176881, −4.10236233557890925730362171639, −3.92638207136741738388198006209, −3.92236702118636896872016919171, −3.71185010931561430205058939615, −3.43038728551498513091799668575, −3.41613180644578975320913754604, −3.38333471750921419888161323043, −3.35856452007855614460753331635, −3.30615628793293271115707648141, −2.75752939405162850066083874209, −2.62062189297894309263430543161, −2.59477202382168440653913990664, −2.33677087136624328700912409957, −2.01128700507516402033317613058, −1.92956007692562520411985090382, −1.86971751396767937078021084878, −1.69119877555508751673430966393, −1.28412729943793320920310523370, −1.02465140551265902156588079696, −0.805515622695376134455033660239, −0.77840246554160124668860940418, −0.77201311773139268623855979040,
0.77201311773139268623855979040, 0.77840246554160124668860940418, 0.805515622695376134455033660239, 1.02465140551265902156588079696, 1.28412729943793320920310523370, 1.69119877555508751673430966393, 1.86971751396767937078021084878, 1.92956007692562520411985090382, 2.01128700507516402033317613058, 2.33677087136624328700912409957, 2.59477202382168440653913990664, 2.62062189297894309263430543161, 2.75752939405162850066083874209, 3.30615628793293271115707648141, 3.35856452007855614460753331635, 3.38333471750921419888161323043, 3.41613180644578975320913754604, 3.43038728551498513091799668575, 3.71185010931561430205058939615, 3.92236702118636896872016919171, 3.92638207136741738388198006209, 4.10236233557890925730362171639, 4.16178965812161801250090176881, 4.43069021815222524224207791804, 4.65348975009075887704352324318
Plot not available for L-functions of degree greater than 10.
