| L(s) = 1 | + 2-s − 3·4-s − 4·5-s − 5·8-s − 4·10-s − 7·13-s + 8·17-s − 11·19-s + 12·20-s − 5·23-s + 10·25-s − 7·26-s + 17·29-s − 5·31-s + 9·32-s + 8·34-s + 15·37-s − 11·38-s + 20·40-s + 10·41-s + 4·43-s − 5·46-s + 8·47-s − 18·49-s + 10·50-s + 21·52-s − 10·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3/2·4-s − 1.78·5-s − 1.76·8-s − 1.26·10-s − 1.94·13-s + 1.94·17-s − 2.52·19-s + 2.68·20-s − 1.04·23-s + 2·25-s − 1.37·26-s + 3.15·29-s − 0.898·31-s + 1.59·32-s + 1.37·34-s + 2.46·37-s − 1.78·38-s + 3.16·40-s + 1.56·41-s + 0.609·43-s − 0.737·46-s + 1.16·47-s − 2.57·49-s + 1.41·50-s + 2.91·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_4\times C_2$ | \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.2.ab_e_ac_j |
| 7 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 18 T^{2} - 15 T^{3} + 149 T^{4} - 15 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_s_ap_ft |
| 13 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 7 T + 36 T^{2} + 71 T^{3} + 239 T^{4} + 71 p T^{5} + 36 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.h_bk_ct_jf |
| 17 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 62 T^{2} - 320 T^{3} + 1471 T^{4} - 320 p T^{5} + 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ai_ck_ami_cep |
| 19 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 11 T + 82 T^{2} + 443 T^{3} + 2155 T^{4} + 443 p T^{5} + 82 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.l_de_rb_dex |
| 23 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 5 T + 67 T^{2} + 370 T^{3} + 2019 T^{4} + 370 p T^{5} + 67 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.f_cp_og_czr |
| 29 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 17 T + 200 T^{2} - 1517 T^{3} + 9499 T^{4} - 1517 p T^{5} + 200 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ar_hs_acgj_obj |
| 31 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 5 T + 79 T^{2} + 230 T^{3} + 2971 T^{4} + 230 p T^{5} + 79 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.f_db_iw_ekh |
| 37 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 15 T + 213 T^{2} - 1710 T^{3} + 12869 T^{4} - 1710 p T^{5} + 213 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ap_if_acnu_taz |
| 41 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 10 T + 154 T^{2} - 1085 T^{3} + 9261 T^{4} - 1085 p T^{5} + 154 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ak_fy_abpt_nsf |
| 43 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 4 T + 48 T^{2} + 25 T^{3} - 139 T^{4} + 25 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_bw_z_afj |
| 47 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 172 T^{2} - 985 T^{3} + 11871 T^{4} - 985 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ai_gq_ablx_rop |
| 53 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 10 T + 202 T^{2} + 1565 T^{3} + 15819 T^{4} + 1565 p T^{5} + 202 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.k_hu_cif_xkl |
| 59 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 9 T + 167 T^{2} + 1332 T^{3} + 14085 T^{4} + 1332 p T^{5} + 167 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.j_gl_bzg_uvt |
| 61 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 37 T + 748 T^{2} + 9769 T^{3} + 90385 T^{4} + 9769 p T^{5} + 748 p^{2} T^{6} + 37 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.bl_bcu_olt_fdsj |
| 67 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 3 T + 222 T^{2} - 675 T^{3} + 20801 T^{4} - 675 p T^{5} + 222 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ad_io_azz_beub |
| 71 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 13 T + 323 T^{2} + 2686 T^{3} + 35395 T^{4} + 2686 p T^{5} + 323 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.n_ml_dzi_cajj |
| 73 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 15 T + 252 T^{2} + 1965 T^{3} + 21929 T^{4} + 1965 p T^{5} + 252 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.p_js_cxp_bgll |
| 79 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 20 T + 411 T^{2} + 4750 T^{3} + 52151 T^{4} + 4750 p T^{5} + 411 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.u_pv_has_czdv |
| 83 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 17 T + 71 T^{2} - 1114 T^{3} - 16661 T^{4} - 1114 p T^{5} + 71 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.r_ct_abqw_ayqv |
| 89 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 24 T + 227 T^{2} - 342 T^{3} - 7305 T^{4} - 342 p T^{5} + 227 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ay_it_ane_akuz |
| 97 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 32 T + 672 T^{2} + 9775 T^{3} + 111611 T^{4} + 9775 p T^{5} + 672 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bg_zw_olz_gjct |
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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
−6.11830192630218940601423839697, −5.95828516275282201680745203024, −5.79514087766208283700243334798, −5.51838382719726925497542715288, −5.34494794761161000344670151522, −4.80742385419375415021874270023, −4.75538351397966103282954366297, −4.72960134880894814624193316341, −4.63432691216895937211209647398, −4.35904926646242435184730230047, −4.28691314635808408554297563472, −4.24627654522234103887825971431, −4.01873641578071365334442774205, −3.48882514223633204625024118623, −3.41691474722920636026880721295, −3.34010239121811198066748957902, −2.89989513778854017256550470701, −2.83698121218055226659217874541, −2.58504120668626241017197342334, −2.45499662723772761412229954444, −2.26667735913714858937615770585, −1.51968044233397277249694638386, −1.49160557502398683336521644400, −1.06634269930936756412298839729, −0.954840425050713038033483549134, 0, 0, 0, 0,
0.954840425050713038033483549134, 1.06634269930936756412298839729, 1.49160557502398683336521644400, 1.51968044233397277249694638386, 2.26667735913714858937615770585, 2.45499662723772761412229954444, 2.58504120668626241017197342334, 2.83698121218055226659217874541, 2.89989513778854017256550470701, 3.34010239121811198066748957902, 3.41691474722920636026880721295, 3.48882514223633204625024118623, 4.01873641578071365334442774205, 4.24627654522234103887825971431, 4.28691314635808408554297563472, 4.35904926646242435184730230047, 4.63432691216895937211209647398, 4.72960134880894814624193316341, 4.75538351397966103282954366297, 4.80742385419375415021874270023, 5.34494794761161000344670151522, 5.51838382719726925497542715288, 5.79514087766208283700243334798, 5.95828516275282201680745203024, 6.11830192630218940601423839697
