| L(s) = 1 | + 8·7-s + 8·13-s − 16·23-s − 8·25-s + 8·31-s + 8·37-s − 16·47-s + 20·49-s + 16·59-s + 24·61-s + 16·67-s − 16·71-s − 8·73-s + 24·79-s + 8·89-s + 64·91-s − 16·97-s + 16·101-s + 24·103-s + 16·107-s + 24·109-s + 8·113-s − 28·121-s + 16·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 2.21·13-s − 3.33·23-s − 8/5·25-s + 1.43·31-s + 1.31·37-s − 2.33·47-s + 20/7·49-s + 2.08·59-s + 3.07·61-s + 1.95·67-s − 1.89·71-s − 0.936·73-s + 2.70·79-s + 0.847·89-s + 6.70·91-s − 1.62·97-s + 1.59·101-s + 2.36·103-s + 1.54·107-s + 2.29·109-s + 0.752·113-s − 2.54·121-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.92520365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.92520365\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| good | 5 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T^{2} - 16 T^{3} + 26 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_i_aq_ba |
| 7 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 44 T^{2} - 24 p T^{3} + 510 T^{4} - 24 p^{2} T^{5} + 44 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ai_bs_agm_tq |
| 11 | $C_2^2:C_4$ | \( 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_bc_a_pq |
| 13 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 40 T^{2} - 136 T^{3} + 514 T^{4} - 136 p T^{5} + 40 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ai_bo_afg_tu |
| 17 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 36 T^{2} + 64 T^{3} + 614 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_bk_cm_xq |
| 19 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 44 T^{2} - 64 T^{3} + 918 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_bs_acm_bji |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.23.q_hg_cai_lpu |
| 29 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 72 T^{2} - 112 T^{3} + 2426 T^{4} - 112 p T^{5} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_cu_aei_dpi |
| 31 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 108 T^{2} - 616 T^{3} + 162 p T^{4} - 616 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ai_ee_axs_hle |
| 37 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 104 T^{2} - 776 T^{3} + 5346 T^{4} - 776 p T^{5} + 104 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_ea_abdw_hxq |
| 41 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 68 T^{2} - 64 T^{3} + 3206 T^{4} - 64 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_cq_acm_eti |
| 43 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 76 T^{2} + 64 T^{3} + 3830 T^{4} + 64 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_cy_cm_fri |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.q_im_cyy_yli |
| 53 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 104 T^{2} - 272 T^{3} + 5594 T^{4} - 272 p T^{5} + 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ea_akm_ihe |
| 59 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 16 T + 204 T^{2} - 1552 T^{3} + 12758 T^{4} - 1552 p T^{5} + 204 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.aq_hw_achs_sws |
| 61 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 24 T + 392 T^{2} - 4568 T^{3} + 40386 T^{4} - 4568 p T^{5} + 392 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ay_pc_agts_chti |
| 67 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 16 T + 172 T^{2} - 912 T^{3} + 6134 T^{4} - 912 p T^{5} + 172 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.aq_gq_abjc_jby |
| 71 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 16 T + 4 p T^{2} + 2640 T^{3} + 28070 T^{4} + 2640 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.q_ky_dxo_bpnq |
| 73 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T + 172 T^{2} + 1720 T^{3} + 16006 T^{4} + 1720 p T^{5} + 172 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.i_gq_coe_xrq |
| 79 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 24 T + 396 T^{2} - 4344 T^{3} + 42398 T^{4} - 4344 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ay_pg_aglc_ckss |
| 83 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 252 T^{2} + 128 T^{3} + 28598 T^{4} + 128 p T^{5} + 252 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_js_ey_bqhy |
| 89 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 188 T^{2} - 2424 T^{3} + 17894 T^{4} - 2424 p T^{5} + 188 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ai_hg_adpg_bamg |
| 97 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 16 T + 356 T^{2} + 4400 T^{3} + 49990 T^{4} + 4400 p T^{5} + 356 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.q_ns_gng_cvys |
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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
−5.60645196072471490719779400736, −5.15862660169896280398393238805, −4.93818542188093410228966966691, −4.85666281770811325164981559373, −4.78129239628405645477586963268, −4.32233571003576739422521784552, −4.30938509230656936678095743999, −4.20373462910854671649122350459, −4.14092675074401816023401618332, −3.65723603878940972240866022417, −3.50056765148879145111325283009, −3.46666469377130962932680451951, −3.44411546708491957562188775606, −2.84249248328985699291233684837, −2.74717556756050260173470424649, −2.21849956317966545519629383546, −2.20264634548139265953049386588, −2.02648373642045043147482073353, −1.76625240669543843611725175060, −1.70791799525923622734495665147, −1.59596133706480409364298775101, −1.15662112741829394285216021232, −0.66064978597280234918998529697, −0.62900031308888376797520129706, −0.57499884741130546717190939202,
0.57499884741130546717190939202, 0.62900031308888376797520129706, 0.66064978597280234918998529697, 1.15662112741829394285216021232, 1.59596133706480409364298775101, 1.70791799525923622734495665147, 1.76625240669543843611725175060, 2.02648373642045043147482073353, 2.20264634548139265953049386588, 2.21849956317966545519629383546, 2.74717556756050260173470424649, 2.84249248328985699291233684837, 3.44411546708491957562188775606, 3.46666469377130962932680451951, 3.50056765148879145111325283009, 3.65723603878940972240866022417, 4.14092675074401816023401618332, 4.20373462910854671649122350459, 4.30938509230656936678095743999, 4.32233571003576739422521784552, 4.78129239628405645477586963268, 4.85666281770811325164981559373, 4.93818542188093410228966966691, 5.15862660169896280398393238805, 5.60645196072471490719779400736
