| L(s) = 1 | − 6·3-s − 2·5-s − 4·7-s + 16·9-s + 4·11-s − 14·13-s + 12·15-s − 12·17-s − 14·19-s + 24·21-s − 14·23-s + 2·25-s − 24·27-s − 2·29-s − 4·31-s − 24·33-s + 8·35-s − 4·37-s + 84·39-s + 12·41-s − 16·43-s − 32·45-s − 4·47-s − 4·49-s + 72·51-s − 8·55-s + 84·57-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 0.894·5-s − 1.51·7-s + 16/3·9-s + 1.20·11-s − 3.88·13-s + 3.09·15-s − 2.91·17-s − 3.21·19-s + 5.23·21-s − 2.91·23-s + 2/5·25-s − 4.61·27-s − 0.371·29-s − 0.718·31-s − 4.17·33-s + 1.35·35-s − 0.657·37-s + 13.4·39-s + 1.87·41-s − 2.43·43-s − 4.77·45-s − 0.583·47-s − 4/7·49-s + 10.0·51-s − 1.07·55-s + 11.1·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) | |
| good | 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 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 8 T^{3} + 31 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.c_c_i_bf |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 60 T^{3} + 191 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.e_u_ci_hj |
| 11 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} + 56 T^{3} - 233 T^{4} + 56 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ae_i_ce_aiz |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 5 p T^{2} + 444 T^{3} + 1896 T^{4} + 444 p T^{5} + 5 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.m_dh_rc_cuy |
| 19 | $D_4\times C_2$ | \( 1 + 14 T + 74 T^{2} + 144 T^{3} + 47 T^{4} + 144 p T^{5} + 74 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.o_cw_fo_bv |
| 23 | $D_4\times C_2$ | \( 1 + 14 T + 104 T^{2} + 28 p T^{3} + 149 p T^{4} + 28 p^{2} T^{5} + 104 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.o_ea_yu_fbv |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 22 T^{3} + 1252 T^{4} - 22 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.c_abr_aw_bwe |
| 31 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 36 T^{3} - 322 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.e_i_bk_amk |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 29 T^{2} - 288 T^{3} - 988 T^{4} - 288 p T^{5} + 29 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_bd_alc_abma |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 117 T^{2} - 768 T^{3} + 5228 T^{4} - 768 p T^{5} + 117 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.am_en_abdo_htc |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 118 T^{2} + 832 T^{3} + 6187 T^{4} + 832 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.q_eo_bga_jdz |
| 47 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 100 T^{3} + 766 T^{4} + 100 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.e_i_dw_bdm |
| 53 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_hy_a_yad |
| 59 | $D_4\times C_2$ | \( 1 - 16 T + 164 T^{2} - 1276 T^{3} + 9607 T^{4} - 1276 p T^{5} + 164 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.aq_gi_abxc_ofn |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 284 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.e_abj_aky_adcq |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 90 T^{2} - 768 T^{3} - 13297 T^{4} - 768 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.s_dm_abdo_atrl |
| 71 | $C_2^3$ | \( 1 - 14 T + 98 T^{2} + 616 T^{3} - 9353 T^{4} + 616 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ao_du_xs_anvt |
| 73 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} - 264 T^{3} - 9817 T^{4} - 264 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.g_s_ake_aonp |
| 79 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 22854 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_aim_a_bhva |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 124 T^{3} - 782 T^{4} - 124 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ae_i_aeu_abec |
| 89 | $D_4\times C_2$ | \( 1 - 34 T + 458 T^{2} - 2980 T^{3} + 16159 T^{4} - 2980 p T^{5} + 458 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.abi_rq_aekq_xxn |
| 97 | $D_4\times C_2$ | \( 1 + 6 T + 90 T^{2} - 1140 T^{3} - 3889 T^{4} - 1140 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.g_dm_abrw_aftp |
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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
−8.773762275511057013132161773634, −8.239677984262716554362074474739, −8.083833588041360360526623529924, −7.76365953767460892623086339590, −7.52907828636107883479272953311, −7.05945315702305923659983559612, −6.86097421576329633870568688410, −6.72003641261365639287000716936, −6.68183426176098543294024016023, −6.25962163839389954739969504100, −6.16565044475187646255582378982, −6.08311331259063115208410822078, −5.65472154556633669563973750849, −5.33539785964698695858726984153, −4.91971319415928519579151419806, −4.85063132936456542449801648206, −4.55824275627068615391565688342, −4.38964724661892121830375143731, −4.05332683666184781322720996721, −3.71784344794083070911084319261, −3.56799147763417166540814661825, −2.51620608739338730400883558044, −2.43621696157377917817995203036, −2.19750340878796152087120097767, −1.69349809029698303493755643569, 0, 0, 0, 0,
1.69349809029698303493755643569, 2.19750340878796152087120097767, 2.43621696157377917817995203036, 2.51620608739338730400883558044, 3.56799147763417166540814661825, 3.71784344794083070911084319261, 4.05332683666184781322720996721, 4.38964724661892121830375143731, 4.55824275627068615391565688342, 4.85063132936456542449801648206, 4.91971319415928519579151419806, 5.33539785964698695858726984153, 5.65472154556633669563973750849, 6.08311331259063115208410822078, 6.16565044475187646255582378982, 6.25962163839389954739969504100, 6.68183426176098543294024016023, 6.72003641261365639287000716936, 6.86097421576329633870568688410, 7.05945315702305923659983559612, 7.52907828636107883479272953311, 7.76365953767460892623086339590, 8.083833588041360360526623529924, 8.239677984262716554362074474739, 8.773762275511057013132161773634
