| L(s) = 1 | + 4·2-s + 10·4-s − 5·5-s − 7-s + 20·8-s − 20·10-s − 7·11-s + 4·13-s − 4·14-s + 35·16-s − 4·17-s + 2·19-s − 50·20-s − 28·22-s − 16·23-s + 2·25-s + 16·26-s − 10·28-s − 31-s + 56·32-s − 16·34-s + 5·35-s − 2·37-s + 8·38-s − 100·40-s + 41-s + 9·43-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 2.23·5-s − 0.377·7-s + 7.07·8-s − 6.32·10-s − 2.11·11-s + 1.10·13-s − 1.06·14-s + 35/4·16-s − 0.970·17-s + 0.458·19-s − 11.1·20-s − 5.96·22-s − 3.33·23-s + 2/5·25-s + 3.13·26-s − 1.88·28-s − 0.179·31-s + 9.89·32-s − 2.74·34-s + 0.845·35-s − 0.328·37-s + 1.29·38-s − 15.8·40-s + 0.156·41-s + 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 269^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 269^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 3 | | \( 1 \) | |
| 269 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 5 | $((C_8 : C_2):C_2):C_2$ | \( 1 + p T + 23 T^{2} + 69 T^{3} + 176 T^{4} + 69 p T^{5} + 23 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.5.f_x_cr_gu |
| 7 | $((C_8 : C_2):C_2):C_2$ | \( 1 + T + 22 T^{2} + 20 T^{3} + 211 T^{4} + 20 p T^{5} + 22 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.7.b_w_u_id |
| 11 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 7 T + 56 T^{2} + 228 T^{3} + 977 T^{4} + 228 p T^{5} + 56 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.h_ce_iu_blp |
| 13 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 4 T + 41 T^{2} - 126 T^{3} + 60 p T^{4} - 126 p T^{5} + 41 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_bp_aew_bea |
| 17 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 4 T + 23 T^{2} + 72 T^{3} + 188 T^{4} + 72 p T^{5} + 23 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_x_cu_hg |
| 19 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 2 T + 52 T^{2} - 106 T^{3} + 1270 T^{4} - 106 p T^{5} + 52 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ac_ca_aec_bww |
| 23 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 16 T + 171 T^{2} + 1224 T^{3} + 6860 T^{4} + 1224 p T^{5} + 171 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.q_gp_bvc_kdw |
| 29 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 65 T^{2} + 136 T^{3} + 2105 T^{4} + 136 p T^{5} + 65 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_cn_fg_dcz |
| 31 | $((C_8 : C_2):C_2):C_2$ | \( 1 + T + 84 T^{2} + 194 T^{3} + 3219 T^{4} + 194 p T^{5} + 84 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.31.b_dg_hm_etv |
| 37 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 2 T + 107 T^{2} + 44 T^{3} + 4992 T^{4} + 44 p T^{5} + 107 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.c_ed_bs_hka |
| 41 | $((C_8 : C_2):C_2):C_2$ | \( 1 - T + 141 T^{2} - 105 T^{3} + 8252 T^{4} - 105 p T^{5} + 141 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ab_fl_aeb_mfk |
| 43 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 9 T + 94 T^{2} - 704 T^{3} + 6323 T^{4} - 704 p T^{5} + 94 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.aj_dq_abbc_jjf |
| 47 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 13 T + 177 T^{2} + 1285 T^{3} + 11048 T^{4} + 1285 p T^{5} + 177 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.n_gv_bxl_qiy |
| 53 | $D_{4}$ | \( ( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.bc_se_hxs_cpfq |
| 59 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 19 T + 297 T^{2} + 3049 T^{3} + 27896 T^{4} + 3049 p T^{5} + 297 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.t_ll_enh_bpgy |
| 61 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 20 T + 377 T^{2} + 3990 T^{3} + 38820 T^{4} + 3990 p T^{5} + 377 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.u_on_fxm_cflc |
| 67 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 15 T + 346 T^{2} - 3176 T^{3} + 37487 T^{4} - 3176 p T^{5} + 346 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ap_ni_aese_cdlv |
| 71 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 15 T + 243 T^{2} + 2285 T^{3} + 22196 T^{4} + 2285 p T^{5} + 243 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.p_jj_djx_bgvs |
| 73 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 7 T + 134 T^{2} + 918 T^{3} + 15387 T^{4} + 918 p T^{5} + 134 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.h_fe_bji_wtv |
| 79 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 17 T + 333 T^{2} + 3859 T^{3} + 40200 T^{4} + 3859 p T^{5} + 333 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.r_mv_fsl_chme |
| 83 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 6 T + 48 T^{2} - 490 T^{3} - 4786 T^{4} - 490 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.g_bw_asw_ahcc |
| 89 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 20 T + 421 T^{2} + 5296 T^{3} + 58905 T^{4} + 5296 p T^{5} + 421 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.u_qf_hvs_djdp |
| 97 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 3 T + 45 T^{2} - 183 T^{3} + 17108 T^{4} - 183 p T^{5} + 45 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ad_bt_ahb_zia |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.11306501352643377136427449982, −5.87340548123197964253993474890, −5.81417581909194964830498611760, −5.67621298850110808062172856317, −5.57168156613098975120967312145, −5.02252731347650487282577582111, −4.90843977609283384152038567091, −4.74441659112986400700392678630, −4.71619503866090108458334869783, −4.35464538829995459541825289178, −4.29307029482091240396776632351, −4.06866841335358461737234071468, −4.05249859642138101544088343614, −3.46429307075975193027164019098, −3.42285749406430796454312685294, −3.38187938590004151974213757888, −3.38010748996299001221417468532, −2.90398085125589247080758767835, −2.57732236074449970259088925085, −2.56158949180880960128703069395, −2.37665896911330037950070684799, −1.71061380998651325793237859232, −1.63621803093884332331980158473, −1.55447494543000927108292122236, −1.33074331612920739516929959683, 0, 0, 0, 0,
1.33074331612920739516929959683, 1.55447494543000927108292122236, 1.63621803093884332331980158473, 1.71061380998651325793237859232, 2.37665896911330037950070684799, 2.56158949180880960128703069395, 2.57732236074449970259088925085, 2.90398085125589247080758767835, 3.38010748996299001221417468532, 3.38187938590004151974213757888, 3.42285749406430796454312685294, 3.46429307075975193027164019098, 4.05249859642138101544088343614, 4.06866841335358461737234071468, 4.29307029482091240396776632351, 4.35464538829995459541825289178, 4.71619503866090108458334869783, 4.74441659112986400700392678630, 4.90843977609283384152038567091, 5.02252731347650487282577582111, 5.57168156613098975120967312145, 5.67621298850110808062172856317, 5.81417581909194964830498611760, 5.87340548123197964253993474890, 6.11306501352643377136427449982
