| L(s) = 1 | − 2-s − 2·3-s − 3·4-s + 2·6-s − 6·7-s + 8-s − 3·9-s − 9·11-s + 6·12-s − 6·13-s + 6·14-s + 7·16-s − 9·17-s + 3·18-s + 12·21-s + 9·22-s − 4·23-s − 2·24-s + 6·26-s + 13·27-s + 18·28-s + 3·29-s − 3·31-s − 2·32-s + 18·33-s + 9·34-s + 9·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 3/2·4-s + 0.816·6-s − 2.26·7-s + 0.353·8-s − 9-s − 2.71·11-s + 1.73·12-s − 1.66·13-s + 1.60·14-s + 7/4·16-s − 2.18·17-s + 0.707·18-s + 2.61·21-s + 1.91·22-s − 0.834·23-s − 0.408·24-s + 1.17·26-s + 2.50·27-s + 3.40·28-s + 0.557·29-s − 0.538·31-s − 0.353·32-s + 3.13·33-s + 1.54·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 5 | \( 1 \) | |
| 7 | \( ( 1 + T )^{6} \) | |
| 13 | \( ( 1 + T )^{6} \) | |
| good | 2 | \( 1 + T + p^{2} T^{2} + 3 p T^{3} + 5 p T^{4} + 19 T^{5} + 9 p T^{6} + 19 p T^{7} + 5 p^{3} T^{8} + 3 p^{4} T^{9} + p^{6} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | 6.2.b_e_g_k_t_s |
| 3 | \( 1 + 2 T + 7 T^{2} + 7 T^{3} + 28 T^{4} + 35 T^{5} + 116 T^{6} + 35 p T^{7} + 28 p^{2} T^{8} + 7 p^{3} T^{9} + 7 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.3.c_h_h_bc_bj_em |
| 11 | \( 1 + 9 T + 7 p T^{2} + 417 T^{3} + 2097 T^{4} + 8223 T^{5} + 30210 T^{6} + 8223 p T^{7} + 2097 p^{2} T^{8} + 417 p^{3} T^{9} + 7 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.j_cz_qb_dcr_meh_bsry |
| 17 | \( 1 + 9 T + 98 T^{2} + 600 T^{3} + 3999 T^{4} + 18435 T^{5} + 89124 T^{6} + 18435 p T^{7} + 3999 p^{2} T^{8} + 600 p^{3} T^{9} + 98 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.j_du_xc_fxv_bbhb_fbvw |
| 19 | \( 1 + 45 T^{2} + 31 T^{3} + 785 T^{4} + 1819 T^{5} + 11018 T^{6} + 1819 p T^{7} + 785 p^{2} T^{8} + 31 p^{3} T^{9} + 45 p^{4} T^{10} + p^{6} T^{12} \) | 6.19.a_bt_bf_bef_crz_qhu |
| 23 | \( 1 + 4 T + 50 T^{2} + 9 T^{3} + 951 T^{4} - 1556 T^{5} + 25767 T^{6} - 1556 p T^{7} + 951 p^{2} T^{8} + 9 p^{3} T^{9} + 50 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.e_by_j_bkp_achw_bmdb |
| 29 | \( 1 - 3 T + 128 T^{2} - 378 T^{3} + 7571 T^{4} - 20625 T^{5} + 272525 T^{6} - 20625 p T^{7} + 7571 p^{2} T^{8} - 378 p^{3} T^{9} + 128 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.ad_ey_aoo_lff_abenh_pndt |
| 31 | \( 1 + 3 T + 63 T^{2} + 306 T^{3} + 2693 T^{4} + 11479 T^{5} + 102430 T^{6} + 11479 p T^{7} + 2693 p^{2} T^{8} + 306 p^{3} T^{9} + 63 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.d_cl_lu_dzp_qzn_fvnq |
| 37 | \( 1 - 9 T + 178 T^{2} - 1059 T^{3} + 12720 T^{4} - 1529 p T^{5} + 555160 T^{6} - 1529 p^{2} T^{7} + 12720 p^{2} T^{8} - 1059 p^{3} T^{9} + 178 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.aj_gw_abot_svg_adfrx_bfpgi |
| 41 | \( 1 + 10 T + 157 T^{2} + 1279 T^{3} + 10935 T^{4} + 73699 T^{5} + 506486 T^{6} + 73699 p T^{7} + 10935 p^{2} T^{8} + 1279 p^{3} T^{9} + 157 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.k_gb_bxf_qep_efap_bcvgg |
| 43 | \( 1 + 7 T + 103 T^{2} + 646 T^{3} + 8561 T^{4} + 41435 T^{5} + 379878 T^{6} + 41435 p T^{7} + 8561 p^{2} T^{8} + 646 p^{3} T^{9} + 103 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.h_dz_yw_mrh_cjhr_vpys |
| 47 | \( 1 - 3 T + 150 T^{2} - 556 T^{3} + 11471 T^{4} - 48441 T^{5} + 620484 T^{6} - 48441 p T^{7} + 11471 p^{2} T^{8} - 556 p^{3} T^{9} + 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.ad_fu_avk_qzf_actrd_bjhwu |
| 53 | \( 1 + 34 T + 673 T^{2} + 9205 T^{3} + 99354 T^{4} + 885745 T^{5} + 6893960 T^{6} + 885745 p T^{7} + 99354 p^{2} T^{8} + 9205 p^{3} T^{9} + 673 p^{4} T^{10} + 34 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.bi_zx_nqb_fqzi_bykhd_pcgei |
| 59 | \( 1 - 18 T + 363 T^{2} - 4735 T^{3} + 54557 T^{4} - 525009 T^{5} + 4321254 T^{6} - 525009 p T^{7} + 54557 p^{2} T^{8} - 4735 p^{3} T^{9} + 363 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.as_nz_ahad_dcsj_abdwqr_jlwkc |
| 61 | \( 1 + 6 T + 211 T^{2} + 1259 T^{3} + 25388 T^{4} + 131889 T^{5} + 1869696 T^{6} + 131889 p T^{7} + 25388 p^{2} T^{8} + 1259 p^{3} T^{9} + 211 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.g_id_bwl_blom_hncr_ecjvk |
| 67 | \( 1 - 30 T + 740 T^{2} - 11809 T^{3} + 162064 T^{4} - 1703772 T^{5} + 15675056 T^{6} - 1703772 p T^{7} + 162064 p^{2} T^{8} - 11809 p^{3} T^{9} + 740 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.abe_bcm_armf_jftg_adsyjs_bihvyu |
| 71 | \( 1 + 25 T + 431 T^{2} + 5997 T^{3} + 72041 T^{4} + 733915 T^{5} + 6585800 T^{6} + 733915 p T^{7} + 72041 p^{2} T^{8} + 5997 p^{3} T^{9} + 431 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.z_qp_iwr_ecov_bptrn_oksia |
| 73 | \( 1 - 8 T + 155 T^{2} - 571 T^{3} + 14475 T^{4} - 79121 T^{5} + 1460642 T^{6} - 79121 p T^{7} + 14475 p^{2} T^{8} - 571 p^{3} T^{9} + 155 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.ai_fz_avz_vkt_aenbd_dfcso |
| 79 | \( 1 + 14 T + 497 T^{2} + 5150 T^{3} + 99266 T^{4} + 782862 T^{5} + 10469265 T^{6} + 782862 p T^{7} + 99266 p^{2} T^{8} + 5150 p^{3} T^{9} + 497 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.o_td_hqc_fqvy_bsocc_wxrcb |
| 83 | \( 1 + 3 T + 284 T^{2} + 570 T^{3} + 37533 T^{4} + 51423 T^{5} + 3459660 T^{6} + 51423 p T^{7} + 37533 p^{2} T^{8} + 570 p^{3} T^{9} + 284 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.d_ky_vy_cdnp_cybv_hovvw |
| 89 | \( 1 - 9 T + 243 T^{2} - 2048 T^{3} + 34517 T^{4} - 222087 T^{5} + 3489414 T^{6} - 222087 p T^{7} + 34517 p^{2} T^{8} - 2048 p^{3} T^{9} + 243 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.aj_jj_adau_bzbp_amqnv_hqnwg |
| 97 | \( 1 + 33 T + 696 T^{2} + 9926 T^{3} + 107795 T^{4} + 998285 T^{5} + 9142736 T^{6} + 998285 p T^{7} + 107795 p^{2} T^{8} + 9926 p^{3} T^{9} + 696 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.bh_bau_oru_gdlz_ceutp_uaets |
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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
−5.27608318339709332802442930737, −4.96619521187474663857278745745, −4.91575860692598273473156041225, −4.65164368139369922363342303646, −4.64997906628152304156021228745, −4.50662026160972066300423857677, −4.37467083002692643751081247206, −4.03092380493181762078694132977, −3.91651415760222657585094136009, −3.82931995201549427569767071003, −3.63512018845341927191169479876, −3.47897661140699146690325135497, −3.08504240826902237438095875453, −3.04035008426388588827050171328, −2.98752480855989513499665013450, −2.93619655747743901286537484586, −2.61197235922041628247136940884, −2.46628581242647603756142937985, −2.37664918369303415575380890449, −2.12242075868211349248164595892, −2.11055766003328957392923536537, −1.74306625209109335741197101249, −1.20283539314753327514474469330, −1.00918744823386892307723802939, −0.961269476124225048812219442482, 0, 0, 0, 0, 0, 0,
0.961269476124225048812219442482, 1.00918744823386892307723802939, 1.20283539314753327514474469330, 1.74306625209109335741197101249, 2.11055766003328957392923536537, 2.12242075868211349248164595892, 2.37664918369303415575380890449, 2.46628581242647603756142937985, 2.61197235922041628247136940884, 2.93619655747743901286537484586, 2.98752480855989513499665013450, 3.04035008426388588827050171328, 3.08504240826902237438095875453, 3.47897661140699146690325135497, 3.63512018845341927191169479876, 3.82931995201549427569767071003, 3.91651415760222657585094136009, 4.03092380493181762078694132977, 4.37467083002692643751081247206, 4.50662026160972066300423857677, 4.64997906628152304156021228745, 4.65164368139369922363342303646, 4.91575860692598273473156041225, 4.96619521187474663857278745745, 5.27608318339709332802442930737
Plot not available for L-functions of degree greater than 10.
