| L(s) = 1 | − 2·3-s − 4·7-s − 9-s − 3·11-s − 4·13-s − 3·17-s + 19-s + 8·21-s − 3·23-s + 2·27-s + 15·29-s − 10·31-s + 6·33-s − 4·37-s + 8·39-s − 9·41-s − 5·43-s − 18·47-s + 5·49-s + 6·51-s − 53-s − 2·57-s + 6·59-s − 7·61-s + 4·63-s − 25·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s − 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.727·17-s + 0.229·19-s + 1.74·21-s − 0.625·23-s + 0.384·27-s + 2.78·29-s − 1.79·31-s + 1.04·33-s − 0.657·37-s + 1.28·39-s − 1.40·41-s − 0.762·43-s − 2.62·47-s + 5/7·49-s + 0.840·51-s − 0.137·53-s − 0.264·57-s + 0.781·59-s − 0.896·61-s + 0.503·63-s − 3.05·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $D_{4}$ | \( ( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.c_f_k_bc |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 11 T^{2} + 6 p T^{3} + 128 T^{4} + 6 p^{2} T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.e_l_bq_ey |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 27 T^{2} + 102 T^{3} + 370 T^{4} + 102 p T^{5} + 27 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.d_bb_dy_og |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 37 T^{2} + 50 T^{3} + 682 T^{4} + 50 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.d_bl_by_bag |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 24 T^{2} + 35 T^{3} + 26 p T^{4} + 35 p T^{5} + 24 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ab_y_bj_ta |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 44 T^{2} + 291 T^{3} + 934 T^{4} + 291 p T^{5} + 44 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.d_bs_lf_bjy |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 131 T^{2} - 798 T^{3} + 4384 T^{4} - 798 p T^{5} + 131 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ap_fb_abes_gmq |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 75 T^{2} + 526 T^{3} + 3332 T^{4} + 526 p T^{5} + 75 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.k_cx_ug_eye |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.e_eq_oq_jmc |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 32 T^{2} + 339 T^{3} + 4126 T^{4} + 339 p T^{5} + 32 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.j_bg_nb_gcs |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 5 T + 96 T^{2} + 705 T^{3} + 4702 T^{4} + 705 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.f_ds_bbd_gyw |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 18 T + 223 T^{2} + 2090 T^{3} + 16240 T^{4} + 2090 p T^{5} + 223 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.s_ip_dck_yaq |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 161 T^{2} - 26 T^{3} + 11282 T^{4} - 26 p T^{5} + 161 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.53.b_gf_aba_qry |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 127 T^{2} - 266 T^{3} + 7000 T^{4} - 266 p T^{5} + 127 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ag_ex_akg_kjg |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 211 T^{2} + 1282 T^{3} + 18308 T^{4} + 1282 p T^{5} + 211 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.h_id_bxi_bbce |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 25 T + 391 T^{2} + 4042 T^{3} + 36278 T^{4} + 4042 p T^{5} + 391 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.z_pb_fzm_cbri |
| 71 | $D_{4}$ | \( ( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bk_bci_oto_fptm |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 13 T + 286 T^{2} + 2731 T^{3} + 31106 T^{4} + 2731 p T^{5} + 286 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.n_la_ebb_buak |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 268 T^{2} + 795 T^{3} + 29830 T^{4} + 795 p T^{5} + 268 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.d_ki_bep_bsdi |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 149 T^{2} - 1018 T^{3} + 11584 T^{4} - 1018 p T^{5} + 149 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ab_ft_abne_rdo |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 27 T + 450 T^{2} - 5769 T^{3} + 61058 T^{4} - 5769 p T^{5} + 450 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.abb_ri_ainx_dmik |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 288 T^{2} - 2710 T^{3} + 38270 T^{4} - 2710 p T^{5} + 288 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ak_lc_aeag_cepy |
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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
−6.68517741630782453845577828848, −6.25530460825869845510884894270, −6.23936422350366396878146817196, −6.16828678159744438046573123054, −6.15142951319748269239088857324, −5.53849230308007101101685390528, −5.44702813546955649777435489938, −5.40439996026484181685990662594, −5.13070760902077541198880262959, −4.84447840615175030074361816410, −4.68026566380887406558507016566, −4.54782909534205783674562721470, −4.24903742668097969715188318745, −4.14789690259649216793268833971, −3.50589800910644013973895021376, −3.43355742239795755406382475358, −3.36621697728578388299458134148, −2.91935175377299268077429128364, −2.86466533400632201945565112618, −2.67174757149069014639367209731, −2.37587863957343796872196550542, −1.87219016062919480569172507471, −1.73956599755209077000373427943, −1.36447526922832608255906075670, −1.06396781838331617664607072972, 0, 0, 0, 0,
1.06396781838331617664607072972, 1.36447526922832608255906075670, 1.73956599755209077000373427943, 1.87219016062919480569172507471, 2.37587863957343796872196550542, 2.67174757149069014639367209731, 2.86466533400632201945565112618, 2.91935175377299268077429128364, 3.36621697728578388299458134148, 3.43355742239795755406382475358, 3.50589800910644013973895021376, 4.14789690259649216793268833971, 4.24903742668097969715188318745, 4.54782909534205783674562721470, 4.68026566380887406558507016566, 4.84447840615175030074361816410, 5.13070760902077541198880262959, 5.40439996026484181685990662594, 5.44702813546955649777435489938, 5.53849230308007101101685390528, 6.15142951319748269239088857324, 6.16828678159744438046573123054, 6.23936422350366396878146817196, 6.25530460825869845510884894270, 6.68517741630782453845577828848
