| L(s) = 1 | + 2·3-s + 6·5-s − 10·7-s + 3·9-s + 11·11-s + 4·13-s + 12·15-s − 6·17-s + 9·19-s − 20·21-s − 12·23-s + 25·25-s + 10·27-s + 6·29-s + 6·31-s + 22·33-s − 60·35-s + 2·37-s + 8·39-s − 18·41-s − 6·43-s + 18·45-s − 8·47-s + 47·49-s − 12·51-s + 12·53-s + 66·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.68·5-s − 3.77·7-s + 9-s + 3.31·11-s + 1.10·13-s + 3.09·15-s − 1.45·17-s + 2.06·19-s − 4.36·21-s − 2.50·23-s + 5·25-s + 1.92·27-s + 1.11·29-s + 1.07·31-s + 3.82·33-s − 10.1·35-s + 0.328·37-s + 1.28·39-s − 2.81·41-s − 0.914·43-s + 2.68·45-s − 1.16·47-s + 47/7·49-s − 1.68·51-s + 1.64·53-s + 8.89·55-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
| \(L(1)\) |
\(\approx\) |
\(7.544862300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.544862300\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - p T + 51 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2:C_4$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + 19 T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_b_ag_t |
| 5 | $C_2^2:C_4$ | \( 1 - 6 T + 11 T^{2} - 6 T^{3} + T^{4} - 6 p T^{5} + 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ag_l_ag_b |
| 7 | $C_4\times C_2$ | \( 1 + 10 T + 53 T^{2} + 200 T^{3} + 589 T^{4} + 200 p T^{5} + 53 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.k_cb_hs_wr |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_d_aby_nd |
| 17 | $C_2^2:C_4$ | \( 1 + 6 T + 19 T^{2} + 132 T^{3} + 829 T^{4} + 132 p T^{5} + 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.g_t_fc_bfx |
| 19 | $C_4\times C_2$ | \( 1 - 9 T + 27 T^{2} + 83 T^{3} - 960 T^{4} + 83 p T^{5} + 27 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.aj_bb_df_abky |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.m_fg_bhs_hso |
| 29 | $C_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 288 T^{3} + 2365 T^{4} - 288 p T^{5} + 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ag_bv_alc_dmz |
| 31 | $C_2^2:C_4$ | \( 1 - 6 T + 45 T^{2} - 304 T^{3} + 2589 T^{4} - 304 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ag_bt_als_dvp |
| 37 | $C_2^2:C_4$ | \( 1 - 2 T + 27 T^{2} - 10 T^{3} + 641 T^{4} - 10 p T^{5} + 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ac_bb_ak_yr |
| 41 | $C_2^2:C_4$ | \( 1 + 18 T + 143 T^{2} + 996 T^{3} + 7165 T^{4} + 996 p T^{5} + 143 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.s_fn_bmi_kpp |
| 43 | $D_{4}$ | \( ( 1 + 3 T + 57 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.g_et_xc_lkz |
| 47 | $C_2^2:C_4$ | \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 380 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.i_r_oq_gzp |
| 53 | $C_2^2:C_4$ | \( 1 - 12 T + 11 T^{2} + 84 T^{3} + 1369 T^{4} + 84 p T^{5} + 11 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.am_l_dg_car |
| 59 | $C_2^2:C_4$ | \( 1 + 9 T + 47 T^{2} + 717 T^{3} + 8680 T^{4} + 717 p T^{5} + 47 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.j_bv_bbp_mvw |
| 61 | $C_2^2:C_4$ | \( 1 - 8 T + 3 T^{2} - 436 T^{3} + 6905 T^{4} - 436 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ai_d_aqu_kfp |
| 67 | $D_{4}$ | \( ( 1 + 5 T + 139 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.k_lr_dbg_buvh |
| 71 | $C_2^2:C_4$ | \( 1 - 12 T + 73 T^{2} - 1074 T^{3} + 14005 T^{4} - 1074 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.am_cv_abpi_usr |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 23 T^{2} + 100 T^{3} + 5221 T^{4} + 100 p T^{5} + 23 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.o_x_dw_hsv |
| 79 | $C_4\times C_2$ | \( 1 + 6 T - 43 T^{2} - 732 T^{3} - 995 T^{4} - 732 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.g_abr_abce_abmh |
| 83 | $C_2^2:C_4$ | \( 1 - T - 67 T^{2} + 515 T^{3} + 5516 T^{4} + 515 p T^{5} - 67 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ab_acp_tv_iee |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 233 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.abe_bap_aoho_ggzt |
| 97 | $C_2^2:C_4$ | \( 1 + 7 T - 73 T^{2} - 835 T^{3} + 1816 T^{4} - 835 p T^{5} - 73 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.h_acv_abgd_crw |
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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
−7.48732783249413413681834054727, −6.82911954530770590894586564869, −6.81822511373797338322910416162, −6.78573528335976383938109034313, −6.61617813112791055341457545475, −6.29278662746272501185674873660, −6.27977890288780784862174488771, −6.04047432216168025070274477833, −5.98913506178140364093126176009, −5.40213004293912504244550310794, −5.29353143375326335789786402237, −4.74106400049688524365688333305, −4.58171189544295710916732966794, −4.14134348181334549612609186396, −3.91925101093231083250084616744, −3.67572992401195018250963758290, −3.28685451400596716350181987654, −3.16180234357258718247424922380, −2.93966171085230218847036550709, −2.83295722573001467069187432453, −2.15523052349985498949555398285, −1.84594963936586847451142294416, −1.62604433140085487376753331468, −1.15714577087612049684271118709, −0.71345072790081862459442231268,
0.71345072790081862459442231268, 1.15714577087612049684271118709, 1.62604433140085487376753331468, 1.84594963936586847451142294416, 2.15523052349985498949555398285, 2.83295722573001467069187432453, 2.93966171085230218847036550709, 3.16180234357258718247424922380, 3.28685451400596716350181987654, 3.67572992401195018250963758290, 3.91925101093231083250084616744, 4.14134348181334549612609186396, 4.58171189544295710916732966794, 4.74106400049688524365688333305, 5.29353143375326335789786402237, 5.40213004293912504244550310794, 5.98913506178140364093126176009, 6.04047432216168025070274477833, 6.27977890288780784862174488771, 6.29278662746272501185674873660, 6.61617813112791055341457545475, 6.78573528335976383938109034313, 6.81822511373797338322910416162, 6.82911954530770590894586564869, 7.48732783249413413681834054727
