| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 3·8-s − 4·9-s + 4·11-s − 12-s + 3·13-s − 14-s + 4·16-s − 5·17-s − 4·18-s + 6·19-s + 21-s + 4·22-s − 4·23-s − 3·24-s + 3·26-s + 7·27-s − 28-s + 2·29-s + 21·31-s + 2·32-s − 4·33-s − 5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s − 4/3·9-s + 1.20·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s + 16-s − 1.21·17-s − 0.942·18-s + 1.37·19-s + 0.218·21-s + 0.852·22-s − 0.834·23-s − 0.612·24-s + 0.588·26-s + 1.34·27-s − 0.188·28-s + 0.371·29-s + 3.77·31-s + 0.353·32-s − 0.696·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.701423033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.701423033\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 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 |
| 3 | $C_2 \wr S_5$ | \( 1 + T + 5 T^{2} + 2 T^{3} + 23 T^{4} + 19 T^{5} + 23 p T^{6} + 2 p^{2} T^{7} + 5 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.3.b_f_c_x_t |
| 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.b_n_ba_fh_gn |
| 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.ae_bh_ace_mq_ahw |
| 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.ad_bn_adu_bgv_acpt |
| 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.ac_cr_ahc_eqw_aklw |
| 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.ac_eb_abo_iuc_abyi |
| 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.i_hd_bsa_vec_dvbk |
| 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.ae_db_ahk_hdi_ayhs |
| 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.am_gx_acia_ycs_agkdc |
| 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.am_gd_acro_zss_ajezg |
| 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.av_mz_afdq_bybp_apjaj |
| 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.w_lp_eei_bmco_mlce |
| 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.k_ht_dlw_blvy_nqdg |
| 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.au_uv_akeq_fzag_accbwi |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.61733015120311192739497671982, −6.60943714804173446178976861842, −6.52136385484807824788455224299, −6.25624254535781492607334806829, −6.19824789833829410697511042837, −6.02983365902527318815063093902, −5.56390979334298150860786091297, −5.31735459504638279298642685826, −5.19883094514560799234661131307, −5.08260335648645955696449783511, −4.83713158263332715692091285772, −4.45426399222189972236940654820, −4.33516982401709700523998025002, −4.11341002380124242175242076680, −3.92917325756505295924134791008, −3.48382147393427179616127341770, −3.20351669578278578364384614033, −3.19616275514473665401667350554, −2.90885584951866486531698674038, −2.46975049614380243423524933164, −2.25703778911600787632433421835, −1.81067540079271052228576889822, −1.45571908535926902852269873139, −0.989009460232569672358177642769, −0.62031877100082228789894680633,
0.62031877100082228789894680633, 0.989009460232569672358177642769, 1.45571908535926902852269873139, 1.81067540079271052228576889822, 2.25703778911600787632433421835, 2.46975049614380243423524933164, 2.90885584951866486531698674038, 3.19616275514473665401667350554, 3.20351669578278578364384614033, 3.48382147393427179616127341770, 3.92917325756505295924134791008, 4.11341002380124242175242076680, 4.33516982401709700523998025002, 4.45426399222189972236940654820, 4.83713158263332715692091285772, 5.08260335648645955696449783511, 5.19883094514560799234661131307, 5.31735459504638279298642685826, 5.56390979334298150860786091297, 6.02983365902527318815063093902, 6.19824789833829410697511042837, 6.25624254535781492607334806829, 6.52136385484807824788455224299, 6.60943714804173446178976861842, 6.61733015120311192739497671982
