| L(s) = 1 | − 2·3-s − 5-s − 4·7-s − 4·11-s + 4·13-s + 2·15-s − 11·19-s + 8·21-s − 3·23-s − 14·25-s − 29-s − 31-s + 8·33-s + 4·35-s + 6·37-s − 8·39-s + 8·41-s − 19·43-s + 11·47-s + 10·49-s − 17·53-s + 4·55-s + 22·57-s − 6·59-s + 16·61-s − 4·65-s − 18·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.51·7-s − 1.20·11-s + 1.10·13-s + 0.516·15-s − 2.52·19-s + 1.74·21-s − 0.625·23-s − 2.79·25-s − 0.185·29-s − 0.179·31-s + 1.39·33-s + 0.676·35-s + 0.986·37-s − 1.28·39-s + 1.24·41-s − 2.89·43-s + 1.60·47-s + 10/7·49-s − 2.33·53-s + 0.539·55-s + 2.91·57-s − 0.781·59-s + 2.04·61-s − 0.496·65-s − 2.19·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 2 T + 4 T^{2} + 8 T^{3} + 22 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.c_e_i_w |
| 5 | $C_2 \wr S_4$ | \( 1 + T + 3 p T^{2} + 14 T^{3} + 102 T^{4} + 14 p T^{5} + 3 p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.5.b_p_o_dy |
| 11 | $C_2 \wr S_4$ | \( 1 + 4 T + 26 T^{2} + 86 T^{3} + 442 T^{4} + 86 p T^{5} + 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.e_ba_di_ra |
| 17 | $C_2 \wr S_4$ | \( 1 + 24 T^{2} + 130 T^{3} + 10 p T^{4} + 130 p T^{5} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_y_fa_go |
| 19 | $C_2 \wr S_4$ | \( 1 + 11 T + 81 T^{2} + 484 T^{3} + 2484 T^{4} + 484 p T^{5} + 81 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.l_dd_sq_dro |
| 23 | $C_2 \wr S_4$ | \( 1 + 3 T + 47 T^{2} + 128 T^{3} + 1616 T^{4} + 128 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.d_bv_ey_cke |
| 29 | $C_2 \wr S_4$ | \( 1 + T + 101 T^{2} + 74 T^{3} + 4210 T^{4} + 74 p T^{5} + 101 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.29.b_dx_cw_gfy |
| 31 | $C_2 \wr S_4$ | \( 1 + T + 43 T^{2} + 42 T^{3} + 2000 T^{4} + 42 p T^{5} + 43 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.31.b_br_bq_cyy |
| 37 | $C_2 \wr S_4$ | \( 1 - 6 T + 154 T^{2} - 656 T^{3} + 8654 T^{4} - 656 p T^{5} + 154 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ag_fy_azg_muw |
| 41 | $Q_8:S_4$ | \( 1 - 8 T + 108 T^{2} - 408 T^{3} + 4358 T^{4} - 408 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ai_ee_aps_glq |
| 43 | $C_2 \wr S_4$ | \( 1 + 19 T + 263 T^{2} + 2416 T^{3} + 18456 T^{4} + 2416 p T^{5} + 263 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.t_kd_doy_bbhw |
| 47 | $C_2 \wr S_4$ | \( 1 - 11 T + 189 T^{2} - 1428 T^{3} + 13292 T^{4} - 1428 p T^{5} + 189 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.al_hh_accy_trg |
| 53 | $C_2 \wr S_4$ | \( 1 + 17 T + 305 T^{2} + 2898 T^{3} + 26834 T^{4} + 2898 p T^{5} + 305 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.r_lt_ehm_bnsc |
| 59 | $C_2 \wr S_4$ | \( 1 + 6 T + 220 T^{2} + 978 T^{3} + 18934 T^{4} + 978 p T^{5} + 220 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.g_im_blq_bcag |
| 61 | $C_2 \wr S_4$ | \( 1 - 16 T + 284 T^{2} - 2768 T^{3} + 26710 T^{4} - 2768 p T^{5} + 284 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aq_ky_aecm_bnni |
| 67 | $C_2 \wr S_4$ | \( 1 + 18 T + 368 T^{2} + 3802 T^{3} + 40398 T^{4} + 3802 p T^{5} + 368 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.s_oe_fqg_chtu |
| 71 | $C_2 \wr S_4$ | \( 1 + 4 T + 266 T^{2} + 806 T^{3} + 27802 T^{4} + 806 p T^{5} + 266 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.e_kg_bfa_bpdi |
| 73 | $C_2 \wr S_4$ | \( 1 - 11 T + 217 T^{2} - 2004 T^{3} + 21698 T^{4} - 2004 p T^{5} + 217 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.al_ij_aczc_bgco |
| 79 | $C_2 \wr S_4$ | \( 1 - 5 T + 227 T^{2} - 720 T^{3} + 23256 T^{4} - 720 p T^{5} + 227 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.af_it_abbs_bikm |
| 83 | $C_2 \wr S_4$ | \( 1 + 19 T + 303 T^{2} + 2686 T^{3} + 27656 T^{4} + 2686 p T^{5} + 303 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.t_lr_dzi_boxs |
| 89 | $C_2 \wr S_4$ | \( 1 + 5 T + 201 T^{2} + 1844 T^{3} + 19490 T^{4} + 1844 p T^{5} + 201 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.f_ht_csy_bcvq |
| 97 | $C_2 \wr S_4$ | \( 1 - 3 T + 369 T^{2} - 888 T^{3} + 52770 T^{4} - 888 p T^{5} + 369 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ad_of_abie_dabq |
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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
−6.37732811667410410501378305439, −6.21498454761168975630677146263, −6.19428335543494767748022390379, −6.14366489233327662376753026202, −5.97409132265245835433730207943, −5.58458022848043272181722981154, −5.40182559482620846650257296256, −5.29478122111230377541974210731, −5.24087362137023529052637137791, −4.68608202871271310991221421796, −4.52869841558829604918045339662, −4.23288980866434993516283993623, −4.18026590106769585705318163662, −3.93270531203035463433743024647, −3.64337030701197396328210886544, −3.61806941749217224924438375188, −3.38149669338834257238890297498, −2.81031051831950932515055301891, −2.74503643337116650804935421903, −2.52313655414947090589295926422, −2.34819899426558407363814489233, −1.82429868177644564203175562422, −1.77360423639140975927343558151, −1.29834736827983640692771185955, −0.994686366863730814988760595905, 0, 0, 0, 0,
0.994686366863730814988760595905, 1.29834736827983640692771185955, 1.77360423639140975927343558151, 1.82429868177644564203175562422, 2.34819899426558407363814489233, 2.52313655414947090589295926422, 2.74503643337116650804935421903, 2.81031051831950932515055301891, 3.38149669338834257238890297498, 3.61806941749217224924438375188, 3.64337030701197396328210886544, 3.93270531203035463433743024647, 4.18026590106769585705318163662, 4.23288980866434993516283993623, 4.52869841558829604918045339662, 4.68608202871271310991221421796, 5.24087362137023529052637137791, 5.29478122111230377541974210731, 5.40182559482620846650257296256, 5.58458022848043272181722981154, 5.97409132265245835433730207943, 6.14366489233327662376753026202, 6.19428335543494767748022390379, 6.21498454761168975630677146263, 6.37732811667410410501378305439
