| L(s) = 1 | + 2·2-s − 4·3-s − 4-s + 4·5-s − 8·6-s − 10·7-s − 4·8-s + 11·9-s + 8·10-s + 2·11-s + 4·12-s − 8·13-s − 20·14-s − 16·15-s + 8·17-s + 22·18-s − 2·19-s − 4·20-s + 40·21-s + 4·22-s + 14·23-s + 16·24-s + 5·25-s − 16·26-s − 20·27-s + 10·28-s − 32·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s − 1/2·4-s + 1.78·5-s − 3.26·6-s − 3.77·7-s − 1.41·8-s + 11/3·9-s + 2.52·10-s + 0.603·11-s + 1.15·12-s − 2.21·13-s − 5.34·14-s − 4.13·15-s + 1.94·17-s + 5.18·18-s − 0.458·19-s − 0.894·20-s + 8.72·21-s + 0.852·22-s + 2.91·23-s + 3.26·24-s + 25-s − 3.13·26-s − 3.84·27-s + 1.88·28-s − 5.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2633071706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2633071706\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + 5 T^{2} - p^{3} T^{3} + 13 T^{4} - p^{4} T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_f_ai_n |
| 3 | $D_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 4 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.e_f_ae_au |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_aq_e_jb |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.i_bg_gm_bfm |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 20 T^{2} + 52 T^{3} - 545 T^{4} + 52 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ai_u_ca_auz |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 4 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.c_abg_ae_bhb |
| 23 | $D_4\times C_2$ | \( 1 - 14 T + 53 T^{2} + 226 T^{3} - 2552 T^{4} + 226 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ao_cb_is_adue |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_adu_a_gbb |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.m_ec_bau_ftr |
| 37 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.m_cu_lc_blv |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_aes_a_jvv |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.g_s_ci_abif |
| 47 | $D_4\times C_2$ | \( 1 - 6 T + 90 T^{2} - 672 T^{3} + 5159 T^{4} - 672 p T^{5} + 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ag_dm_azw_hql |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) | 4.53.ak_by_vo_aiht |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.g_acm_aee_gnb |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.m_gb_byi_qkm |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 1224 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ai_fh_abvc_raa |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.am_mu_adxc_ccww |
| 73 | $D_4\times C_2$ | \( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 600 p T^{5} + 144 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_fo_xc_qgt |
| 79 | $D_4\times C_2$ | \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.g_fs_bfk_tnv |
| 83 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 140 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ac_c_afk_ogl |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) | 4.89.aq_dl_gu_aeei |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ae_i_am_anbe |
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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
−12.60282445755374996299588715303, −12.36707297959856234496540541752, −12.35867052808607800355457882741, −11.97194184050155289138667914683, −11.39537090770584928811341159329, −10.64131038979286408187379748068, −10.44769742647340497977959157926, −10.18578176480774244802181954191, −10.11112204958812509586472806943, −9.605355614521322375568214136091, −9.254401039858783429129051177157, −9.191541056020698248138915215520, −9.016155391805571685312311114409, −7.55526143247604282137241975552, −7.03649622037917151838553307701, −7.02713094948501020345620315248, −6.64196071871561793304760132804, −6.12742042860604732342413657475, −5.87517417431976660515400118012, −5.25648486983864906788879731340, −5.15100185885921508299932878107, −5.02892437117064729044505224360, −3.94232016056919743188316307347, −3.62902032686266143692343504868, −2.87831884721365919588530969877,
2.87831884721365919588530969877, 3.62902032686266143692343504868, 3.94232016056919743188316307347, 5.02892437117064729044505224360, 5.15100185885921508299932878107, 5.25648486983864906788879731340, 5.87517417431976660515400118012, 6.12742042860604732342413657475, 6.64196071871561793304760132804, 7.02713094948501020345620315248, 7.03649622037917151838553307701, 7.55526143247604282137241975552, 9.016155391805571685312311114409, 9.191541056020698248138915215520, 9.254401039858783429129051177157, 9.605355614521322375568214136091, 10.11112204958812509586472806943, 10.18578176480774244802181954191, 10.44769742647340497977959157926, 10.64131038979286408187379748068, 11.39537090770584928811341159329, 11.97194184050155289138667914683, 12.35867052808607800355457882741, 12.36707297959856234496540541752, 12.60282445755374996299588715303
