| L(s) = 1 | − 2-s + 3·5-s + 4·7-s − 3·10-s + 2·11-s + 4·13-s − 4·14-s − 16-s + 2·17-s + 7·19-s − 2·22-s − 3·23-s − 25-s − 4·26-s − 29-s + 3·31-s + 32-s − 2·34-s + 12·35-s + 10·37-s − 7·38-s + 16·41-s + 3·43-s + 3·46-s − 5·47-s + 10·49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s + 1.51·7-s − 0.948·10-s + 0.603·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s + 1.60·19-s − 0.426·22-s − 0.625·23-s − 1/5·25-s − 0.784·26-s − 0.185·29-s + 0.538·31-s + 0.176·32-s − 0.342·34-s + 2.02·35-s + 1.64·37-s − 1.13·38-s + 2.49·41-s + 0.457·43-s + 0.442·46-s − 0.729·47-s + 10/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.937833786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.937833786\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + T^{2} + T^{3} + p T^{4} + p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.2.b_b_b_c |
| 5 | $C_2 \wr S_4$ | \( 1 - 3 T + 2 p T^{2} - p^{2} T^{3} + 74 T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ad_k_az_cw |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 20 T^{2} - 34 T^{3} + 294 T^{4} - 34 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_u_abi_li |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 40 T^{2} - 62 T^{3} + 878 T^{4} - 62 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ac_bo_ack_bhu |
| 19 | $C_2 \wr S_4$ | \( 1 - 7 T + 64 T^{2} - 351 T^{3} + 1774 T^{4} - 351 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ah_cm_ann_cqg |
| 23 | $C_2 \wr S_4$ | \( 1 + 3 T + 40 T^{2} - 49 T^{3} + 494 T^{4} - 49 p T^{5} + 40 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.d_bo_abx_ta |
| 29 | $C_2 \wr S_4$ | \( 1 + T + 86 T^{2} + 35 T^{3} + 3378 T^{4} + 35 p T^{5} + 86 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.29.b_di_bj_ezy |
| 31 | $C_2 \wr S_4$ | \( 1 - 3 T - 4 T^{2} - 119 T^{3} + 58 p T^{4} - 119 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ad_ae_aep_cre |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 270 T^{3} + 1870 T^{4} - 270 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ak_cm_akk_cty |
| 41 | $C_2 \wr S_4$ | \( 1 - 16 T + 4 p T^{2} - 1280 T^{3} + 8694 T^{4} - 1280 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.aq_gi_abxg_mwk |
| 43 | $C_2 \wr S_4$ | \( 1 - 3 T + 128 T^{2} - 275 T^{3} + 7246 T^{4} - 275 p T^{5} + 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ad_ey_akp_kss |
| 47 | $C_2 \wr S_4$ | \( 1 + 5 T + 148 T^{2} + 689 T^{3} + 9638 T^{4} + 689 p T^{5} + 148 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.f_fs_ban_ogs |
| 53 | $C_2 \wr S_4$ | \( 1 + 5 T + 174 T^{2} + 727 T^{3} + 12802 T^{4} + 727 p T^{5} + 174 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.f_gs_bbz_syk |
| 59 | $C_2 \wr S_4$ | \( 1 - 20 T + 316 T^{2} - 3236 T^{3} + 28790 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.au_me_aeum_bqpi |
| 61 | $C_2 \wr S_4$ | \( 1 - 12 T + 180 T^{2} - 1508 T^{3} + 15014 T^{4} - 1508 p T^{5} + 180 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.am_gy_acga_wfm |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 228 T^{2} + 1254 T^{3} + 6086 T^{4} + 1254 p T^{5} + 228 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.w_iu_bwg_jac |
| 71 | $C_2 \wr S_4$ | \( 1 + 52 T^{2} - 304 T^{3} + 7478 T^{4} - 304 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ca_als_lbq |
| 73 | $C_2 \wr S_4$ | \( 1 + 13 T + 126 T^{2} - 261 T^{3} - 3934 T^{4} - 261 p T^{5} + 126 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.n_ew_akb_afvi |
| 79 | $C_2 \wr S_4$ | \( 1 - 11 T + 196 T^{2} - 1167 T^{3} + 15030 T^{4} - 1167 p T^{5} + 196 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.al_ho_absx_wgc |
| 83 | $C_2 \wr S_4$ | \( 1 + T + 296 T^{2} + 329 T^{3} + 35310 T^{4} + 329 p T^{5} + 296 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.83.b_lk_mr_cagc |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 194 T^{2} - 139 T^{3} + 16986 T^{4} - 139 p T^{5} + 194 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.af_hm_afj_zdi |
| 97 | $C_2 \wr S_4$ | \( 1 + 17 T + 374 T^{2} + 4127 T^{3} + 52210 T^{4} + 4127 p T^{5} + 374 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.r_ok_gct_czgc |
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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
−7.45552810405133247702557969757, −7.31967627776395811483665507546, −6.86283475840592156110528526496, −6.76400944160991136264233991732, −6.22244172744608057430674102766, −6.10072214084671699471370987012, −5.97736731001218146286314080687, −5.79717234614344026682672616112, −5.65922147296462747163548619693, −5.33751356976887636667312193259, −4.89909229473395663804729591941, −4.78869916393380660889032739047, −4.62086474306083946563342037055, −4.10924421135909017115088314113, −3.98792616036477248081099696503, −3.82486102654575584478101461838, −3.30452041657188043684688722046, −3.06693361403093261452552081085, −2.67595658066238511636727158123, −2.35639325876714136462744642525, −1.95596649930847849140520523718, −1.84670722984431796738065311147, −1.32174650167891728778885897141, −0.939250738215150946036081632070, −0.806379825894722431926348053392,
0.806379825894722431926348053392, 0.939250738215150946036081632070, 1.32174650167891728778885897141, 1.84670722984431796738065311147, 1.95596649930847849140520523718, 2.35639325876714136462744642525, 2.67595658066238511636727158123, 3.06693361403093261452552081085, 3.30452041657188043684688722046, 3.82486102654575584478101461838, 3.98792616036477248081099696503, 4.10924421135909017115088314113, 4.62086474306083946563342037055, 4.78869916393380660889032739047, 4.89909229473395663804729591941, 5.33751356976887636667312193259, 5.65922147296462747163548619693, 5.79717234614344026682672616112, 5.97736731001218146286314080687, 6.10072214084671699471370987012, 6.22244172744608057430674102766, 6.76400944160991136264233991732, 6.86283475840592156110528526496, 7.31967627776395811483665507546, 7.45552810405133247702557969757
