| L(s) = 1 | − 6·2-s − 6·3-s + 17·4-s + 36·6-s − 4·7-s − 30·8-s + 16·9-s − 12·11-s − 102·12-s + 2·13-s + 24·14-s + 40·16-s − 6·17-s − 96·18-s − 4·19-s + 24·21-s + 72·22-s + 6·23-s + 180·24-s − 14·25-s − 12·26-s − 24·27-s − 68·28-s − 18·29-s − 16·31-s − 54·32-s + 72·33-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 3.46·3-s + 17/2·4-s + 14.6·6-s − 1.51·7-s − 10.6·8-s + 16/3·9-s − 3.61·11-s − 29.4·12-s + 0.554·13-s + 6.41·14-s + 10·16-s − 1.45·17-s − 22.6·18-s − 0.917·19-s + 5.23·21-s + 15.3·22-s + 1.25·23-s + 36.7·24-s − 2.79·25-s − 2.35·26-s − 4.61·27-s − 12.8·28-s − 3.34·29-s − 2.87·31-s − 9.54·32-s + 12.5·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(409^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(409^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 409 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | |
| good | 2 | $C_2^2$ | \( ( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.2.g_t_bq_cr |
| 3 | $D_4\times C_2$ | \( 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) | 4.3.g_u_bw_dn |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_o_a_dv |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 60 T^{3} + 191 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.e_u_ci_hj |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.m_cu_nk_cag |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 24 T^{3} + 287 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ac_c_ay_lb |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.g_f_as_ci |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 60 T^{3} + 434 T^{4} + 60 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.e_i_ci_qs |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ag_ca_ajg_bzv |
| 29 | $C_2^3$ | \( 1 + 18 T + 162 T^{2} + 972 T^{3} + 5207 T^{4} + 972 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.s_gg_blk_hsh |
| 31 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 912 T^{3} + 5822 T^{4} + 912 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.q_ey_bjc_ipy |
| 37 | $D_4\times C_2$ | \( 1 + 22 T + 290 T^{2} + 2592 T^{3} + 17975 T^{4} + 2592 p T^{5} + 290 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.w_le_dvs_bapj |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.m_ec_bau_fkl |
| 43 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ai_bg_aps_hpq |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 180 T^{2} - 1428 T^{3} + 12623 T^{4} - 1428 p T^{5} + 180 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.am_gy_accy_srn |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 50 T^{2} + 144 T^{3} - 1605 T^{4} + 144 p T^{5} + 50 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.am_by_fo_acjt |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 3000 T^{3} + 26894 T^{4} - 3000 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ay_lc_aelk_bnuk |
| 61 | $D_4\times C_2$ | \( 1 - 26 T + 338 T^{2} - 3744 T^{3} + 34583 T^{4} - 3744 p T^{5} + 338 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aba_na_afoa_bzed |
| 67 | $D_4\times C_2$ | \( 1 + 28 T + 296 T^{2} + 1224 T^{3} + 2183 T^{4} + 1224 p T^{5} + 296 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.bc_lk_bvc_dfz |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.m_abi_qq_yhn |
| 73 | $D_4\times C_2$ | \( 1 - 34 T + 458 T^{2} - 3108 T^{3} + 18047 T^{4} - 3108 p T^{5} + 458 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.abi_rq_aepo_basd |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.au_hs_advg_btny |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.m_oq_ens_cuzq |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.y_si_jbk_dyda |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 18 T + p T^{2} )^{2}( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) | 4.97.bc_hp_adwq_adccq |
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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
−8.532853399004013485496125211271, −8.532233287352073417710675834669, −8.390706831822321560923472979260, −8.088965167745127683422913620832, −7.50163507248096151369724302691, −7.40811203026248311271524808161, −7.20098318228978586570759475418, −7.10447031789049204534864130994, −7.09324414930802075461270773300, −6.33178203434780358885628949868, −6.27323663351687743758373230250, −5.81285345945544085999426569075, −5.66683179681231655092417310126, −5.54452019232569610415928727519, −5.33865597197598272241493987432, −5.13759453904436590228077958397, −4.99837296827155892070528627920, −3.97412778840025849903438783762, −3.82887526263359154591352525412, −3.61784352943595801687011398057, −3.09059570446152166599001616557, −2.24357173464182659785483255557, −2.03565977249798668737482206862, −1.93142442332244212890297891194, −0.869262529923063390704962256273, 0, 0, 0, 0,
0.869262529923063390704962256273, 1.93142442332244212890297891194, 2.03565977249798668737482206862, 2.24357173464182659785483255557, 3.09059570446152166599001616557, 3.61784352943595801687011398057, 3.82887526263359154591352525412, 3.97412778840025849903438783762, 4.99837296827155892070528627920, 5.13759453904436590228077958397, 5.33865597197598272241493987432, 5.54452019232569610415928727519, 5.66683179681231655092417310126, 5.81285345945544085999426569075, 6.27323663351687743758373230250, 6.33178203434780358885628949868, 7.09324414930802075461270773300, 7.10447031789049204534864130994, 7.20098318228978586570759475418, 7.40811203026248311271524808161, 7.50163507248096151369724302691, 8.088965167745127683422913620832, 8.390706831822321560923472979260, 8.532233287352073417710675834669, 8.532853399004013485496125211271
