| L(s) = 1 | + 2-s + 4-s + 7-s + 3·8-s − 4·11-s − 3·13-s + 14-s + 4·16-s − 5·17-s + 6·19-s − 4·22-s − 4·23-s − 3·26-s + 28-s − 2·29-s + 21·31-s + 2·32-s − 5·34-s − 2·37-s + 6·38-s + 8·41-s − 4·43-s − 4·44-s − 4·46-s + 2·47-s − 12·49-s − 3·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.20·11-s − 0.832·13-s + 0.267·14-s + 16-s − 1.21·17-s + 1.37·19-s − 0.852·22-s − 0.834·23-s − 0.588·26-s + 0.188·28-s − 0.371·29-s + 3.77·31-s + 0.353·32-s − 0.857·34-s − 0.328·37-s + 0.973·38-s + 1.24·41-s − 0.609·43-s − 0.603·44-s − 0.589·46-s + 0.291·47-s − 1.71·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.147355869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147355869\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 17 | $C_1$ | \( ( 1 + T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 - T - p T^{3} + T^{4} + 3 T^{5} + p T^{6} - p^{3} T^{7} - p^{4} T^{9} + p^{5} T^{10} \) | 5.2.ab_a_ac_b_d |
| 7 | $C_2 \wr S_5$ | \( 1 - T + 13 T^{2} - 26 T^{3} + 137 T^{4} - 169 T^{5} + 137 p T^{6} - 26 p^{2} T^{7} + 13 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.7.ab_n_aba_fh_agn |
| 11 | $C_2 \wr S_5$ | \( 1 + 4 T + 3 p T^{2} + 56 T^{3} + 328 T^{4} + 204 T^{5} + 328 p T^{6} + 56 p^{2} T^{7} + 3 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.e_bh_ce_mq_hw |
| 13 | $C_2 \wr S_5$ | \( 1 + 3 T + 3 p T^{2} + 98 T^{3} + 853 T^{4} + 1761 T^{5} + 853 p T^{6} + 98 p^{2} T^{7} + 3 p^{4} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.d_bn_du_bgv_cpt |
| 19 | $C_2 \wr S_5$ | \( 1 - 6 T + 63 T^{2} - 8 p T^{3} + 1114 T^{4} - 1044 T^{5} + 1114 p T^{6} - 8 p^{3} T^{7} + 63 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.ag_cl_afw_bqw_aboe |
| 23 | $C_2 \wr S_5$ | \( 1 + 4 T + 3 p T^{2} + 116 T^{3} + 1792 T^{4} + 996 T^{5} + 1792 p T^{6} + 116 p^{2} T^{7} + 3 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.e_cr_em_cqy_bmi |
| 29 | $C_2 \wr S_5$ | \( 1 + 2 T + 69 T^{2} + 184 T^{3} + 3142 T^{4} + 7068 T^{5} + 3142 p T^{6} + 184 p^{2} T^{7} + 69 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.c_cr_hc_eqw_klw |
| 31 | $C_2 \wr S_5$ | \( 1 - 21 T + 257 T^{2} - 2246 T^{3} + 16111 T^{4} - 96739 T^{5} + 16111 p T^{6} - 2246 p^{2} T^{7} + 257 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.av_jx_adik_xvr_afnct |
| 37 | $C_2 \wr S_5$ | \( 1 + 2 T + 105 T^{2} + 40 T^{3} + 5930 T^{4} + 1308 T^{5} + 5930 p T^{6} + 40 p^{2} T^{7} + 105 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.c_eb_bo_iuc_byi |
| 41 | $C_2 \wr S_5$ | \( 1 - 8 T + 185 T^{2} - 1144 T^{3} + 14302 T^{4} - 66960 T^{5} + 14302 p T^{6} - 1144 p^{2} T^{7} + 185 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.ai_hd_absa_vec_advbk |
| 43 | $C_2 \wr S_5$ | \( 1 + 4 T + 79 T^{2} + 192 T^{3} + 4818 T^{4} + 16424 T^{5} + 4818 p T^{6} + 192 p^{2} T^{7} + 79 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.e_db_hk_hdi_yhs |
| 47 | $C_2 \wr S_5$ | \( 1 - 2 T + 147 T^{2} + 104 T^{3} + 9154 T^{4} + 18132 T^{5} + 9154 p T^{6} + 104 p^{2} T^{7} + 147 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.ac_fr_ea_noc_bavk |
| 53 | $C_2 \wr S_5$ | \( 1 + 21 T + 419 T^{2} + 4902 T^{3} + 52981 T^{4} + 401643 T^{5} + 52981 p T^{6} + 4902 p^{2} T^{7} + 419 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.v_qd_hgo_dajt_wwdv |
| 59 | $C_2 \wr S_5$ | \( 1 + 12 T + 179 T^{2} + 1560 T^{3} + 16294 T^{4} + 112296 T^{5} + 16294 p T^{6} + 1560 p^{2} T^{7} + 179 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.m_gx_cia_ycs_gkdc |
| 61 | $C_2 \wr S_5$ | \( 1 + 2 T + 265 T^{2} + 408 T^{3} + 30258 T^{4} + 35692 T^{5} + 30258 p T^{6} + 408 p^{2} T^{7} + 265 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.c_kf_ps_bstu_cauu |
| 67 | $C_2 \wr S_5$ | \( 1 + 12 T + 159 T^{2} + 1808 T^{3} + 17386 T^{4} + 161544 T^{5} + 17386 p T^{6} + 1808 p^{2} T^{7} + 159 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.m_gd_cro_zss_jezg |
| 71 | $C_2 \wr S_5$ | \( 1 + 21 T + 337 T^{2} + 3474 T^{3} + 33841 T^{4} + 269733 T^{5} + 33841 p T^{6} + 3474 p^{2} T^{7} + 337 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.v_mz_fdq_bybp_pjaj |
| 73 | $C_2 \wr S_5$ | \( 1 - 22 T + 301 T^{2} - 2816 T^{3} + 25754 T^{4} - 218404 T^{5} + 25754 p T^{6} - 2816 p^{2} T^{7} + 301 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.aw_lp_aeei_bmco_amlce |
| 79 | $C_2 \wr S_5$ | \( 1 - 41 T + 919 T^{2} - 14404 T^{3} + 175265 T^{4} - 1721495 T^{5} + 175265 p T^{6} - 14404 p^{2} T^{7} + 919 p^{3} T^{8} - 41 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.abp_bjj_avia_jzgz_adtypj |
| 83 | $C_2 \wr S_5$ | \( 1 - 8 T + 351 T^{2} - 2176 T^{3} + 53722 T^{4} - 256944 T^{5} + 53722 p T^{6} - 2176 p^{2} T^{7} + 351 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.ai_nn_adfs_dbmg_aoqcm |
| 89 | $C_2 \wr S_5$ | \( 1 - 10 T + 201 T^{2} - 2336 T^{3} + 25582 T^{4} - 239388 T^{5} + 25582 p T^{6} - 2336 p^{2} T^{7} + 201 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.ak_ht_adlw_blvy_anqdg |
| 97 | $C_2 \wr S_5$ | \( 1 + 20 T + 541 T^{2} + 6880 T^{3} + 104786 T^{4} + 950360 T^{5} + 104786 p T^{6} + 6880 p^{2} T^{7} + 541 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.u_uv_keq_fzag_ccbwi |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.91097216616634626552248134008, −4.78704319720698846659147231987, −4.78395174582309514487993515424, −4.72335102446650808066930619566, −4.54644980408214063232174801641, −4.13372033530547434180257046336, −4.05660754363193088866615148444, −4.01300561961220784810267438344, −3.77054402803082828816321279934, −3.47021416225245983371319614129, −3.15412870984696503832471880860, −3.14880218264278577406321386117, −3.13530955299142793950793230185, −2.77630892940706799329505449812, −2.54227787699616812461864527120, −2.50321683885295313180680324203, −2.28693693615273496784184232721, −1.98263318586716852321741456892, −1.91524310025070788149256959777, −1.55273937598646258608554082136, −1.32863911095622388516489703078, −1.20909895888427919186801672506, −0.926670775622428417568341248357, −0.47101239605874939706575889770, −0.15537406371244558030027732991,
0.15537406371244558030027732991, 0.47101239605874939706575889770, 0.926670775622428417568341248357, 1.20909895888427919186801672506, 1.32863911095622388516489703078, 1.55273937598646258608554082136, 1.91524310025070788149256959777, 1.98263318586716852321741456892, 2.28693693615273496784184232721, 2.50321683885295313180680324203, 2.54227787699616812461864527120, 2.77630892940706799329505449812, 3.13530955299142793950793230185, 3.14880218264278577406321386117, 3.15412870984696503832471880860, 3.47021416225245983371319614129, 3.77054402803082828816321279934, 4.01300561961220784810267438344, 4.05660754363193088866615148444, 4.13372033530547434180257046336, 4.54644980408214063232174801641, 4.72335102446650808066930619566, 4.78395174582309514487993515424, 4.78704319720698846659147231987, 4.91097216616634626552248134008
