| L(s) = 1 | + 3-s − 3·5-s − 7·7-s − 2·9-s + 4·11-s + 13-s − 3·15-s − 17-s − 7·19-s − 7·21-s + 8·23-s + 10·25-s + 5·29-s − 5·31-s + 4·33-s + 21·35-s + 9·37-s + 39-s − 5·41-s + 6·45-s − 5·47-s + 22·49-s − 51-s − 11·53-s − 12·55-s − 7·57-s − 17·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 2.64·7-s − 2/3·9-s + 1.20·11-s + 0.277·13-s − 0.774·15-s − 0.242·17-s − 1.60·19-s − 1.52·21-s + 1.66·23-s + 2·25-s + 0.928·29-s − 0.898·31-s + 0.696·33-s + 3.54·35-s + 1.47·37-s + 0.160·39-s − 0.780·41-s + 0.894·45-s − 0.729·47-s + 22/7·49-s − 0.140·51-s − 1.51·53-s − 1.61·55-s − 0.927·57-s − 2.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.331008449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331008449\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2:C_4$ | \( 1 - T + p T^{2} - 5 T^{3} + 16 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ab_d_af_q |
| 5 | $C_2^2:C_4$ | \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 3 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.d_ab_ad_q |
| 7 | $C_2^2:C_4$ | \( 1 + p T + 27 T^{2} + 95 T^{3} + 296 T^{4} + 95 p T^{5} + 27 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.7.h_bb_dr_lk |
| 13 | $C_2^2:C_4$ | \( 1 - T + 3 T^{2} + 25 T^{3} + 56 T^{4} + 25 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ab_d_z_ce |
| 17 | $C_2^2:C_4$ | \( 1 + T - T^{2} - 53 T^{3} + 104 T^{4} - 53 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.17.b_ab_acb_ea |
| 19 | $C_2^2:C_4$ | \( 1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 107 p T^{5} + 15 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.h_p_ed_bfs |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ai_cy_aqi_dzq |
| 29 | $C_2^2:C_4$ | \( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 115 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.af_bf_ael_ds |
| 31 | $C_2^2:C_4$ | \( 1 + 5 T + 9 T^{2} + 205 T^{3} + 1916 T^{4} + 205 p T^{5} + 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.f_j_hx_cvs |
| 37 | $C_2^2:C_4$ | \( 1 - 9 T - T^{2} + 57 T^{3} + 784 T^{4} + 57 p T^{5} - p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.aj_ab_cf_bee |
| 41 | $C_2^2:C_4$ | \( 1 + 5 T + 19 T^{2} - 65 T^{3} - 624 T^{4} - 65 p T^{5} + 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.f_t_acn_aya |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2^2:C_4$ | \( 1 + 5 T - 37 T^{2} - 5 p T^{3} + 824 T^{4} - 5 p^{2} T^{5} - 37 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.f_abl_ajb_bfs |
| 53 | $C_2^2:C_4$ | \( 1 + 11 T + 43 T^{2} + 565 T^{3} + 6936 T^{4} + 565 p T^{5} + 43 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.l_br_vt_kgu |
| 59 | $C_2^2:C_4$ | \( 1 + 17 T + 55 T^{2} - 443 T^{3} - 4776 T^{4} - 443 p T^{5} + 55 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.r_cd_arb_ahbs |
| 61 | $C_2^2:C_4$ | \( 1 - T - 45 T^{2} + 361 T^{3} + 2744 T^{4} + 361 p T^{5} - 45 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ab_abt_nx_ebo |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.q_hw_dgi_bhfq |
| 71 | $C_2^2:C_4$ | \( 1 - 5 T - 11 T^{2} + 515 T^{3} - 1164 T^{4} + 515 p T^{5} - 11 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.af_al_tv_absu |
| 73 | $C_2^2:C_4$ | \( 1 - 19 T + 63 T^{2} + 1015 T^{3} - 13384 T^{4} + 1015 p T^{5} + 63 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.at_cl_bnb_atuu |
| 79 | $C_2^2:C_4$ | \( 1 - 3 T + 65 T^{2} - 123 T^{3} + 3244 T^{4} - 123 p T^{5} + 65 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ad_cn_aet_euu |
| 83 | $C_2^2:C_4$ | \( 1 - 31 T + 553 T^{2} - 7355 T^{3} + 75636 T^{4} - 7355 p T^{5} + 553 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.abf_vh_akwx_ehxc |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.q_pw_gfw_dhby |
| 97 | $C_2^2:C_4$ | \( 1 - 27 T + 227 T^{2} - 585 T^{3} + 256 T^{4} - 585 p T^{5} + 227 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.abb_it_awn_jw |
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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
−7.58535378819339680378677560041, −7.47406410765642805865835880153, −6.81076851063586642722526889711, −6.67403977309142180190183415680, −6.61890992781271299429149767857, −6.44910202063307187353499655421, −6.24974107636490625514315421361, −5.99498783673428524466660878488, −5.95869080046415823698039639943, −5.11738289459668878902935492869, −5.02672853747684395921581011389, −4.89710558511264514934792430125, −4.51484966660585505930634030663, −4.20214456075464380748814655327, −4.06462190240710091871075370879, −3.43343108604074772348455074479, −3.39753451073728063607618607773, −3.36953520543249603467193659566, −3.08925667266382710451016269368, −2.72874610953053552109588852412, −2.42653311690763419936965979003, −1.93867406905306822173357077615, −1.44102712200036852612189581629, −0.64374974835437508612945788622, −0.49507403361274383421885702279,
0.49507403361274383421885702279, 0.64374974835437508612945788622, 1.44102712200036852612189581629, 1.93867406905306822173357077615, 2.42653311690763419936965979003, 2.72874610953053552109588852412, 3.08925667266382710451016269368, 3.36953520543249603467193659566, 3.39753451073728063607618607773, 3.43343108604074772348455074479, 4.06462190240710091871075370879, 4.20214456075464380748814655327, 4.51484966660585505930634030663, 4.89710558511264514934792430125, 5.02672853747684395921581011389, 5.11738289459668878902935492869, 5.95869080046415823698039639943, 5.99498783673428524466660878488, 6.24974107636490625514315421361, 6.44910202063307187353499655421, 6.61890992781271299429149767857, 6.67403977309142180190183415680, 6.81076851063586642722526889711, 7.47406410765642805865835880153, 7.58535378819339680378677560041
