| L(s) = 1 | − 3·3-s − 5·5-s + 2·9-s − 8·13-s + 15·15-s − 4·17-s + 6·23-s + 4·25-s + 4·27-s + 8·29-s + 7·31-s + 8·37-s + 24·39-s − 15·41-s − 9·43-s − 10·45-s − 6·47-s + 12·51-s + 17·53-s − 5·61-s + 40·65-s − 9·67-s − 18·69-s − 12·71-s + 11·73-s − 12·75-s − 6·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.23·5-s + 2/3·9-s − 2.21·13-s + 3.87·15-s − 0.970·17-s + 1.25·23-s + 4/5·25-s + 0.769·27-s + 1.48·29-s + 1.25·31-s + 1.31·37-s + 3.84·39-s − 2.34·41-s − 1.37·43-s − 1.49·45-s − 0.875·47-s + 1.68·51-s + 2.33·53-s − 0.640·61-s + 4.96·65-s − 1.09·67-s − 2.16·69-s − 1.42·71-s + 1.28·73-s − 1.38·75-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| 17 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 + p T + 7 T^{2} + 11 T^{3} + 16 T^{4} + 11 p T^{5} + 7 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.3.d_h_l_q |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + p T + 21 T^{2} + 67 T^{3} + 156 T^{4} + 67 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.5.f_v_cp_ga |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 142 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_q_a_fm |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.13.i_cy_ng_clq |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.19.a_cy_a_dfi |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 72 T^{2} - 286 T^{3} + 2126 T^{4} - 286 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ag_cu_ala_ddu |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 84 T^{2} - 504 T^{3} + 3110 T^{4} - 504 p T^{5} + 84 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ai_dg_atk_epq |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 21 T^{2} + 229 T^{3} - 2220 T^{4} + 229 p T^{5} + 21 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ah_v_iv_adhk |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 144 T^{2} - 808 T^{3} + 7854 T^{4} - 808 p T^{5} + 144 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_fo_abfc_lqc |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 15 T + 177 T^{2} + 1413 T^{3} + 10020 T^{4} + 1413 p T^{5} + 177 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.p_gv_ccj_ovk |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 117 T^{2} + 561 T^{3} + 5196 T^{4} + 561 p T^{5} + 117 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.j_en_vp_hrw |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 64 T^{2} + 78 T^{3} + 2238 T^{4} + 78 p T^{5} + 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.g_cm_da_dic |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 17 T + 235 T^{2} - 2023 T^{3} + 16792 T^{4} - 2023 p T^{5} + 235 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ar_jb_aczv_yvw |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 124 T^{2} + 8182 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_eu_a_mcs |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 5 T + 245 T^{2} + 907 T^{3} + 22444 T^{4} + 907 p T^{5} + 245 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.f_jl_bix_bhfg |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 215 T^{2} + 1833 T^{3} + 19912 T^{4} + 1833 p T^{5} + 215 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.j_ih_csn_bdlw |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 140 T^{2} + 1148 T^{3} + 7238 T^{4} + 1148 p T^{5} + 140 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.m_fk_bse_ksk |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 193 T^{2} - 1857 T^{3} + 20804 T^{4} - 1857 p T^{5} + 193 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.al_hl_actl_beue |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 192 T^{2} + 654 T^{3} + 18494 T^{4} + 654 p T^{5} + 192 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.g_hk_ze_bbji |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 554 T^{3} + 686 T^{4} + 554 p T^{5} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ag_a_vi_bak |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 180 T^{2} + 1054 T^{3} + 18198 T^{4} + 1054 p T^{5} + 180 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.c_gy_boo_baxy |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 265 T^{2} + 1795 T^{3} + 34052 T^{4} + 1795 p T^{5} + 265 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.j_kf_crb_byjs |
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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
−6.06789849688203944146663285355, −5.55033847530101397768506529121, −5.53809654880268890178515481703, −5.45877974842894115013110593943, −5.29782001735506399410587899049, −4.95543310188566999352361097766, −4.79356072970362911348524132723, −4.77593834805979697086067710139, −4.50210354978781521374780738748, −4.42742172657953523835047500404, −4.06578556214398215920060011206, −4.00905955196596645410886179830, −3.97282636360171543238411411467, −3.32762845392939670139613723938, −3.24203890799963181956548684010, −3.21432642311390062232477621595, −2.98367699893027027697963628006, −2.56583224806059467413537712622, −2.49638516175571115217004105934, −2.15573529183641527372458221676, −2.12617508680991099697810547355, −1.53788503512230636198867666469, −1.26718097695889262145540620961, −0.889536339560773716695598055707, −0.888053066860085782292849344258, 0, 0, 0, 0,
0.888053066860085782292849344258, 0.889536339560773716695598055707, 1.26718097695889262145540620961, 1.53788503512230636198867666469, 2.12617508680991099697810547355, 2.15573529183641527372458221676, 2.49638516175571115217004105934, 2.56583224806059467413537712622, 2.98367699893027027697963628006, 3.21432642311390062232477621595, 3.24203890799963181956548684010, 3.32762845392939670139613723938, 3.97282636360171543238411411467, 4.00905955196596645410886179830, 4.06578556214398215920060011206, 4.42742172657953523835047500404, 4.50210354978781521374780738748, 4.77593834805979697086067710139, 4.79356072970362911348524132723, 4.95543310188566999352361097766, 5.29782001735506399410587899049, 5.45877974842894115013110593943, 5.53809654880268890178515481703, 5.55033847530101397768506529121, 6.06789849688203944146663285355
