| L(s) = 1 | − 6·2-s − 6·3-s + 17·4-s − 6·5-s + 36·6-s − 2·7-s − 30·8-s + 16·9-s + 36·10-s − 12·11-s − 102·12-s − 4·13-s + 12·14-s + 36·15-s + 40·16-s − 12·17-s − 96·18-s − 8·19-s − 102·20-s + 12·21-s + 72·22-s − 12·23-s + 180·24-s + 27·25-s + 24·26-s − 24·27-s − 34·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 3.46·3-s + 17/2·4-s − 2.68·5-s + 14.6·6-s − 0.755·7-s − 10.6·8-s + 16/3·9-s + 11.3·10-s − 3.61·11-s − 29.4·12-s − 1.10·13-s + 3.20·14-s + 9.29·15-s + 10·16-s − 2.91·17-s − 22.6·18-s − 1.83·19-s − 22.8·20-s + 2.61·21-s + 15.3·22-s − 2.50·23-s + 36.7·24-s + 27/5·25-s + 4.70·26-s − 4.61·27-s − 6.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88529281 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88529281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) | |
| good | 2 | $C_2^2$ | \( ( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.2.g_t_bq_cr |
| 3 | $D_4\times C_2$ | \( 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) | 4.3.g_u_bw_dn |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) | 4.5.g_j_abe_afg |
| 7 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.c_c_ay_acv |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.m_de_ps_cid |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 24 T^{3} - 220 T^{4} - 24 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.e_f_ay_aim |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 45 T^{2} - 48 T^{3} - 820 T^{4} - 48 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.m_bt_abw_abfo |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 168 T^{3} + 878 T^{4} + 168 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.i_bg_gm_bhu |
| 23 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} + 312 T^{3} + 1343 T^{4} + 312 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.m_cu_ma_bzr |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 90 T^{2} - 456 T^{3} + 3719 T^{4} - 456 p T^{5} + 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ag_dm_aro_fnb |
| 31 | $D_4\times C_2$ | \( 1 + 18 T + 196 T^{2} + 1584 T^{3} + 10131 T^{4} + 1584 p T^{5} + 196 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.s_ho_ciy_ozr |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) | 4.37.c_dx_ew_gyi |
| 41 | $C_2^3$ | \( 1 + 18 T + 162 T^{2} + 972 T^{3} + 5663 T^{4} + 972 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.s_gg_blk_ijv |
| 43 | $D_4\times C_2$ | \( 1 + 6 T - 32 T^{2} - 108 T^{3} + 1227 T^{4} - 108 p T^{5} - 32 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.g_abg_aee_bvf |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.ay_lw_aepc_bkne |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_dt_a_jtw |
| 59 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} + 288 T^{3} - 73 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.m_cu_lc_acv |
| 61 | $C_2^3$ | \( 1 - 119 T^{2} + 10440 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_aep_a_plo |
| 67 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 252 T^{3} + 7922 T^{4} + 252 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.e_i_js_lss |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 306 T^{2} - 3192 T^{3} + 32567 T^{4} - 3192 p T^{5} + 306 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.as_lu_aesu_bwep |
| 73 | $D_4\times C_2$ | \( 1 + 8 T + 10 T^{2} - 736 T^{3} - 7085 T^{4} - 736 p T^{5} + 10 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.i_k_abci_akmn |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ajk_a_bomo |
| 83 | $D_4\times C_2$ | \( 1 - 18 T + 306 T^{2} - 3336 T^{3} + 36047 T^{4} - 3336 p T^{5} + 306 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.as_lu_aeyi_cbil |
| 89 | $D_4\times C_2$ | \( 1 - 314 T^{2} + 40059 T^{4} - 314 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_amc_a_chgt |
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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
−10.77128648557267204800907376992, −10.44779908660801688124636768210, −10.32820835398359247649851770226, −10.25149301931082784039782435856, −10.21368287055058387198600398772, −9.376351778942095276987034592989, −8.998776468956634929590956354801, −8.814145798117503591717609579872, −8.803224665685721730660145430929, −8.167546737691063710064921623745, −8.021738870439844246617725781368, −7.88228327872724458562663198721, −7.53147628535938555598608560693, −7.33514031388118420733915563574, −6.70487373692994084638934551375, −6.66313573760057585099069640377, −6.43271010030582409414494630439, −5.68453019213349724916531252503, −5.36669962450613449128168717706, −5.15562357750306937119228483213, −4.79395257373395811695236030592, −4.18385563070461793963473214298, −3.83605183194741347107082091241, −2.74058842917760073971498103134, −2.20707893917941011764918952367, 0, 0, 0, 0,
2.20707893917941011764918952367, 2.74058842917760073971498103134, 3.83605183194741347107082091241, 4.18385563070461793963473214298, 4.79395257373395811695236030592, 5.15562357750306937119228483213, 5.36669962450613449128168717706, 5.68453019213349724916531252503, 6.43271010030582409414494630439, 6.66313573760057585099069640377, 6.70487373692994084638934551375, 7.33514031388118420733915563574, 7.53147628535938555598608560693, 7.88228327872724458562663198721, 8.021738870439844246617725781368, 8.167546737691063710064921623745, 8.803224665685721730660145430929, 8.814145798117503591717609579872, 8.998776468956634929590956354801, 9.376351778942095276987034592989, 10.21368287055058387198600398772, 10.25149301931082784039782435856, 10.32820835398359247649851770226, 10.44779908660801688124636768210, 10.77128648557267204800907376992
