| L(s) = 1 | − 3·3-s − 5-s + 2·7-s − 11-s + 5·13-s + 3·15-s + 6·17-s + 2·19-s − 6·21-s − 2·23-s − 8·25-s + 7·27-s − 11·31-s + 3·33-s − 2·35-s + 12·37-s − 15·39-s − 6·41-s − 5·43-s + 13·47-s − 4·49-s − 18·51-s + 3·53-s + 55-s − 6·57-s − 6·59-s − 22·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 0.755·7-s − 0.301·11-s + 1.38·13-s + 0.774·15-s + 1.45·17-s + 0.458·19-s − 1.30·21-s − 0.417·23-s − 8/5·25-s + 1.34·27-s − 1.97·31-s + 0.522·33-s − 0.338·35-s + 1.97·37-s − 2.40·39-s − 0.937·41-s − 0.762·43-s + 1.89·47-s − 4/7·49-s − 2.52·51-s + 0.412·53-s + 0.134·55-s − 0.794·57-s − 0.781·59-s − 2.81·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 29 | | \( 1 \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + p T + p^{2} T^{2} + 20 T^{3} + 38 T^{4} + 20 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.3.d_j_u_bm |
| 5 | $C_2 \wr S_4$ | \( 1 + T + 9 T^{2} + 2 p T^{3} + 62 T^{4} + 2 p^{2} T^{5} + 9 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.5.b_j_k_ck |
| 7 | $C_2 \wr S_4$ | \( 1 - 2 T + 8 T^{2} - 18 T^{3} + 78 T^{4} - 18 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ac_i_as_da |
| 11 | $C_2 \wr S_4$ | \( 1 + T + 25 T^{2} + 56 T^{3} + 306 T^{4} + 56 p T^{5} + 25 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.11.b_z_ce_lu |
| 13 | $C_2 \wr S_4$ | \( 1 - 5 T + 37 T^{2} - 90 T^{3} + 506 T^{4} - 90 p T^{5} + 37 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.af_bl_adm_tm |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 40 T^{2} - 90 T^{3} + 526 T^{4} - 90 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_bo_adm_ug |
| 19 | $C_2 \wr S_4$ | \( 1 - 2 T + 36 T^{2} - 178 T^{3} + 678 T^{4} - 178 p T^{5} + 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ac_bk_agw_bac |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 72 T^{2} + 114 T^{3} + 2318 T^{4} + 114 p T^{5} + 72 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.c_cu_ek_dle |
| 31 | $C_2 \wr S_4$ | \( 1 + 11 T + 125 T^{2} + 856 T^{3} + 5986 T^{4} + 856 p T^{5} + 125 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.l_ev_bgy_iwg |
| 37 | $C_2 \wr S_4$ | \( 1 - 12 T + 116 T^{2} - 948 T^{3} + 6358 T^{4} - 948 p T^{5} + 116 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.am_em_abkm_jko |
| 41 | $C_2 \wr S_4$ | \( 1 + 6 T + 72 T^{2} - 118 T^{3} + 686 T^{4} - 118 p T^{5} + 72 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.g_cu_aeo_bak |
| 43 | $C_2 \wr S_4$ | \( 1 + 5 T + 145 T^{2} + 616 T^{3} + 8890 T^{4} + 616 p T^{5} + 145 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.f_fp_xs_ndy |
| 47 | $C_2 \wr S_4$ | \( 1 - 13 T + 181 T^{2} - 1820 T^{3} + 12598 T^{4} - 1820 p T^{5} + 181 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.an_gz_acsa_sqo |
| 53 | $C_2 \wr S_4$ | \( 1 - 3 T + 157 T^{2} - 558 T^{3} + 11026 T^{4} - 558 p T^{5} + 157 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ad_gb_avm_qic |
| 59 | $C_2 \wr S_4$ | \( 1 + 6 T + 160 T^{2} + 798 T^{3} + 13614 T^{4} + 798 p T^{5} + 160 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.g_ge_bes_udq |
| 61 | $C_2 \wr S_4$ | \( 1 + 22 T + 320 T^{2} + 2978 T^{3} + 25486 T^{4} + 2978 p T^{5} + 320 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.w_mi_eko_blsg |
| 67 | $C_2 \wr S_4$ | \( 1 - 12 T + 140 T^{2} - 1388 T^{3} + 13878 T^{4} - 1388 p T^{5} + 140 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.am_fk_acbk_unu |
| 71 | $C_2 \wr S_4$ | \( 1 + 92 T^{2} + 896 T^{3} + 1958 T^{4} + 896 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_do_bim_cxi |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.73.bg_baa_nki_fghq |
| 79 | $C_2 \wr S_4$ | \( 1 + 17 T + 333 T^{2} + 3244 T^{3} + 37518 T^{4} + 3244 p T^{5} + 333 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.r_mv_euu_cdna |
| 83 | $C_2 \wr S_4$ | \( 1 + 26 T + 384 T^{2} + 3570 T^{3} + 31694 T^{4} + 3570 p T^{5} + 384 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ba_ou_fhi_buxa |
| 89 | $C_2 \wr S_4$ | \( 1 + 26 T + 568 T^{2} + 7606 T^{3} + 85966 T^{4} + 7606 p T^{5} + 568 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ba_vw_lgo_exek |
| 97 | $C_2 \wr S_4$ | \( 1 + 14 T + 280 T^{2} + 2674 T^{3} + 35374 T^{4} + 2674 p T^{5} + 280 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.o_ku_dyw_caio |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.07129768498371065540578637011, −5.64159890315142689468146815881, −5.52742805195800281890285924602, −5.46585490806138241599670404685, −5.35679312583175016455732277165, −5.31925915203736157979785485325, −4.97255400024854047008656017014, −4.80836439836070977122304754554, −4.31681511878469581093470564567, −4.24881059833310997221889347632, −4.01932411320002034130715915250, −4.01553891048254604419146581289, −3.95174348693072529389587518684, −3.57892197127713339764760938860, −3.07541613737483065173651696223, −3.07540400883775411704870180653, −2.99938862749651839021217064995, −2.72049755823808639767249881135, −2.44731808686473607468275604857, −2.16278856096107023396776342680, −1.71850391262091688302677406794, −1.55443165299842049687338237398, −1.23650947401276703741717581277, −1.21131448638531075503496623418, −1.09626088517090802276060007143, 0, 0, 0, 0,
1.09626088517090802276060007143, 1.21131448638531075503496623418, 1.23650947401276703741717581277, 1.55443165299842049687338237398, 1.71850391262091688302677406794, 2.16278856096107023396776342680, 2.44731808686473607468275604857, 2.72049755823808639767249881135, 2.99938862749651839021217064995, 3.07540400883775411704870180653, 3.07541613737483065173651696223, 3.57892197127713339764760938860, 3.95174348693072529389587518684, 4.01553891048254604419146581289, 4.01932411320002034130715915250, 4.24881059833310997221889347632, 4.31681511878469581093470564567, 4.80836439836070977122304754554, 4.97255400024854047008656017014, 5.31925915203736157979785485325, 5.35679312583175016455732277165, 5.46585490806138241599670404685, 5.52742805195800281890285924602, 5.64159890315142689468146815881, 6.07129768498371065540578637011
