| 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
| \(L(1)\) |
\(\approx\) |
\(5.428483547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.428483547\) |
| \(L(\frac{3}{2})\) |
|
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.c_b_g_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.ak_cb_ahs_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.j_bb_adf_abky |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.am_fg_abhs_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.g_bt_ls_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.ag_et_axc_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.ai_r_aoq_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.aj_bv_abbp_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.ak_lr_adbg_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.m_cv_bpi_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.ag_abr_bce_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.b_acp_atv_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
−7.43879931128302334637090420754, −7.32727052179618311240752534430, −7.02171510610364273264067154243, −6.79192909273045539180253645544, −6.61886763197441332899971357358, −6.06544825765021879401704768749, −6.03070855415146616147687034428, −5.71724304250839174734074081493, −5.61406603786829643624574144319, −5.28125678972294138950556970003, −5.15916243391313484853135992600, −4.92530195875929471317692947020, −4.63969724399318743637758225639, −4.63527962887901730486691871934, −4.60767973801875830193780980973, −3.87444004820684997842803619543, −3.54715254853780312555648317694, −2.81353811862754280019985469476, −2.79652239257710338925829962278, −2.18413734609819484813611503590, −2.04445454119361711497729652037, −1.96276890385631457555722941707, −1.74067477481115171052631235867, −1.01852691685013715511785565957, −0.73690053902094785031719316317,
0.73690053902094785031719316317, 1.01852691685013715511785565957, 1.74067477481115171052631235867, 1.96276890385631457555722941707, 2.04445454119361711497729652037, 2.18413734609819484813611503590, 2.79652239257710338925829962278, 2.81353811862754280019985469476, 3.54715254853780312555648317694, 3.87444004820684997842803619543, 4.60767973801875830193780980973, 4.63527962887901730486691871934, 4.63969724399318743637758225639, 4.92530195875929471317692947020, 5.15916243391313484853135992600, 5.28125678972294138950556970003, 5.61406603786829643624574144319, 5.71724304250839174734074081493, 6.03070855415146616147687034428, 6.06544825765021879401704768749, 6.61886763197441332899971357358, 6.79192909273045539180253645544, 7.02171510610364273264067154243, 7.32727052179618311240752534430, 7.43879931128302334637090420754
