| L(s) = 1 | + 4·2-s − 3·3-s + 10·4-s − 7·5-s − 12·6-s + 20·8-s − 2·9-s − 28·10-s + 2·11-s − 30·12-s − 13-s + 21·15-s + 35·16-s − 5·17-s − 8·18-s − 11·19-s − 70·20-s + 8·22-s − 4·23-s − 60·24-s + 16·25-s − 4·26-s + 17·27-s + 2·29-s + 84·30-s − 6·31-s + 56·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.73·3-s + 5·4-s − 3.13·5-s − 4.89·6-s + 7.07·8-s − 2/3·9-s − 8.85·10-s + 0.603·11-s − 8.66·12-s − 0.277·13-s + 5.42·15-s + 35/4·16-s − 1.21·17-s − 1.88·18-s − 2.52·19-s − 15.6·20-s + 1.70·22-s − 0.834·23-s − 12.2·24-s + 16/5·25-s − 0.784·26-s + 3.27·27-s + 0.371·29-s + 15.3·30-s − 1.07·31-s + 9.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 7 | | \( 1 \) | |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + p T + 11 T^{2} + 22 T^{3} + 49 T^{4} + 22 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.3.d_l_w_bx |
| 5 | $C_2 \wr S_4$ | \( 1 + 7 T + 33 T^{2} + 22 p T^{3} + 277 T^{4} + 22 p^{2} T^{5} + 33 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.h_bh_eg_kr |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 32 T^{2} - 37 T^{3} + 453 T^{4} - 37 p T^{5} + 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_bg_abl_rl |
| 13 | $C_2 \wr S_4$ | \( 1 + T + 10 T^{2} + 3 p T^{3} + 265 T^{4} + 3 p^{2} T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.13.b_k_bn_kf |
| 17 | $C_2 \wr S_4$ | \( 1 + 5 T + 32 T^{2} - 11 T^{3} + 87 T^{4} - 11 p T^{5} + 32 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.f_bg_al_dj |
| 19 | $C_2 \wr S_4$ | \( 1 + 11 T + 63 T^{2} + 344 T^{3} + 1773 T^{4} + 344 p T^{5} + 63 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.l_cl_ng_cqf |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 105 T^{2} - 142 T^{3} + 4387 T^{4} - 142 p T^{5} + 105 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ac_eb_afm_gmt |
| 31 | $C_2 \wr S_4$ | \( 1 + 6 T + 58 T^{2} + 97 T^{3} + 967 T^{4} + 97 p T^{5} + 58 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.g_cg_dt_blf |
| 37 | $C_2 \wr S_4$ | \( 1 + 8 T + 88 T^{2} + 225 T^{3} + 2407 T^{4} + 225 p T^{5} + 88 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.i_dk_ir_dop |
| 41 | $C_2 \wr S_4$ | \( 1 + 9 T + 130 T^{2} + 913 T^{3} + 7715 T^{4} + 913 p T^{5} + 130 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.j_fa_bjd_lkt |
| 43 | $C_2 \wr S_4$ | \( 1 - 4 T + 88 T^{2} - 231 T^{3} + 4891 T^{4} - 231 p T^{5} + 88 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_dk_aix_hgd |
| 47 | $C_2 \wr S_4$ | \( 1 + 11 T + 153 T^{2} + 1300 T^{3} + 10651 T^{4} + 1300 p T^{5} + 153 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.l_fx_bya_ptr |
| 53 | $C_2 \wr S_4$ | \( 1 - T + 107 T^{2} + 448 T^{3} + 4791 T^{4} + 448 p T^{5} + 107 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ab_ed_rg_hch |
| 59 | $C_2 \wr S_4$ | \( 1 + 12 T + 254 T^{2} + 2043 T^{3} + 22929 T^{4} + 2043 p T^{5} + 254 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.m_ju_dap_bhxx |
| 61 | $C_2 \wr S_4$ | \( 1 + 21 T + 214 T^{2} + 1147 T^{3} + 6193 T^{4} + 1147 p T^{5} + 214 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.v_ig_bsd_jef |
| 67 | $C_2 \wr S_4$ | \( 1 + 3 T + 66 T^{2} + 163 T^{3} + 5693 T^{4} + 163 p T^{5} + 66 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.d_co_gh_ikz |
| 71 | $C_2 \wr S_4$ | \( 1 - 11 T + 208 T^{2} - 1919 T^{3} + 19265 T^{4} - 1919 p T^{5} + 208 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.al_ia_acvv_bcmz |
| 73 | $C_2 \wr S_4$ | \( 1 - 16 T + 294 T^{2} - 2949 T^{3} + 30773 T^{4} - 2949 p T^{5} + 294 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.aq_li_aejl_btnp |
| 79 | $C_2 \wr S_4$ | \( 1 + 21 T + 418 T^{2} + 4979 T^{3} + 53509 T^{4} + 4979 p T^{5} + 418 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.v_qc_hjn_dbeb |
| 83 | $C_2 \wr S_4$ | \( 1 + 4 T + 177 T^{2} + 644 T^{3} + 21517 T^{4} + 644 p T^{5} + 177 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.e_gv_yu_bfvp |
| 89 | $C_2 \wr S_4$ | \( 1 + 27 T + 493 T^{2} + 6380 T^{3} + 67913 T^{4} + 6380 p T^{5} + 493 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.bb_sz_jlk_dwmb |
| 97 | $C_2 \wr S_4$ | \( 1 + 6 T + 355 T^{2} + 16 p T^{3} + 50029 T^{4} + 16 p^{2} T^{5} + 355 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.g_nr_chs_cwaf |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.81200912212180053107449881470, −6.40398840285060938620696036805, −6.20297510539996548211240649285, −6.19821485634967083884985383190, −6.15493621986919109442691192146, −5.52407644939754820041304529367, −5.48595715182361411907996664351, −5.36776370067646748857922545206, −5.32652554185244990096764643280, −4.71386518617867233010761178731, −4.61488665570332262177510778198, −4.59487145852641761649756827573, −4.31110453197249492877019353848, −4.10770907828184637439194290402, −3.84014737379571252577701075650, −3.73441571583317549324754540437, −3.71167989352987190829935634332, −3.31064562961183925295589136623, −2.89753328676207410165536233012, −2.79616936729187411087471463730, −2.68176928198814297628074075492, −2.05003806605063785597691115905, −2.03234716637821323079326339082, −1.42327246202776324736811793723, −1.35324418924179083541989564013, 0, 0, 0, 0,
1.35324418924179083541989564013, 1.42327246202776324736811793723, 2.03234716637821323079326339082, 2.05003806605063785597691115905, 2.68176928198814297628074075492, 2.79616936729187411087471463730, 2.89753328676207410165536233012, 3.31064562961183925295589136623, 3.71167989352987190829935634332, 3.73441571583317549324754540437, 3.84014737379571252577701075650, 4.10770907828184637439194290402, 4.31110453197249492877019353848, 4.59487145852641761649756827573, 4.61488665570332262177510778198, 4.71386518617867233010761178731, 5.32652554185244990096764643280, 5.36776370067646748857922545206, 5.48595715182361411907996664351, 5.52407644939754820041304529367, 6.15493621986919109442691192146, 6.19821485634967083884985383190, 6.20297510539996548211240649285, 6.40398840285060938620696036805, 6.81200912212180053107449881470
