| 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
| \(L(1)\) |
\(\approx\) |
\(2.307721394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.307721394\) |
| \(L(\frac{3}{2})\) |
|
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.b_d_f_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.ah_bb_adr_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.ah_p_aed_bfs |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.i_cy_qi_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.af_j_ahx_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.af_abl_jb_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.ar_cd_rb_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.aq_hw_adgi_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.f_al_atv_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.d_cn_et_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.bf_vh_kwx_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
−7.40694412271857200217659042069, −7.39717665489389265288166733420, −7.31649116468969426426723184686, −6.89016281666240306454483319165, −6.61574228315089670499584405585, −6.25152969298637410001710633184, −6.04991588466936250634820318579, −5.69163529743477770504957997016, −5.63754266064702976777109146958, −5.30135266675217834385284036289, −5.05880054573175194231920894817, −4.87009189612329166737660587592, −4.49860880451522489033220112844, −4.45020699327486096936462930301, −4.40234277328842936462571588415, −3.64435291031435554192918146688, −3.56745250349664423729822777486, −3.35907847791851430144516477319, −2.67248163844403073051526304940, −2.66510507249567853730934003491, −2.26458136646576152739955297745, −1.78785153857496785201143741674, −1.45609906353880848380137288759, −0.75261745753736474738439418183, −0.63694986575491492808137799990,
0.63694986575491492808137799990, 0.75261745753736474738439418183, 1.45609906353880848380137288759, 1.78785153857496785201143741674, 2.26458136646576152739955297745, 2.66510507249567853730934003491, 2.67248163844403073051526304940, 3.35907847791851430144516477319, 3.56745250349664423729822777486, 3.64435291031435554192918146688, 4.40234277328842936462571588415, 4.45020699327486096936462930301, 4.49860880451522489033220112844, 4.87009189612329166737660587592, 5.05880054573175194231920894817, 5.30135266675217834385284036289, 5.63754266064702976777109146958, 5.69163529743477770504957997016, 6.04991588466936250634820318579, 6.25152969298637410001710633184, 6.61574228315089670499584405585, 6.89016281666240306454483319165, 7.31649116468969426426723184686, 7.39717665489389265288166733420, 7.40694412271857200217659042069
