| L(s) = 1 | − 8·3-s − 5·4-s − 4·5-s − 10·7-s − 8-s + 31·9-s − 8·11-s + 40·12-s − 8·13-s + 32·15-s + 12·16-s + 9·17-s − 16·19-s + 20·20-s + 80·21-s − 23-s + 8·24-s − 25-s − 79·27-s + 50·28-s + 2·29-s + 2·31-s + 7·32-s + 64·33-s + 40·35-s − 155·36-s − 17·37-s + ⋯ |
| L(s) = 1 | − 4.61·3-s − 5/2·4-s − 1.78·5-s − 3.77·7-s − 0.353·8-s + 31/3·9-s − 2.41·11-s + 11.5·12-s − 2.21·13-s + 8.26·15-s + 3·16-s + 2.18·17-s − 3.67·19-s + 4.47·20-s + 17.4·21-s − 0.208·23-s + 1.63·24-s − 1/5·25-s − 15.2·27-s + 9.44·28-s + 0.371·29-s + 0.359·31-s + 1.23·32-s + 11.1·33-s + 6.76·35-s − 25.8·36-s − 2.79·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(197^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 197 | $C_1$ | \( ( 1 + T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + 5 T^{2} + T^{3} + 13 T^{4} + 3 T^{5} + 13 p T^{6} + p^{2} T^{7} + 5 p^{3} T^{8} + p^{5} T^{10} \) | 5.2.a_f_b_n_d |
| 3 | $C_2 \wr S_5$ | \( 1 + 8 T + 11 p T^{2} + 95 T^{3} + 214 T^{4} + 401 T^{5} + 214 p T^{6} + 95 p^{2} T^{7} + 11 p^{4} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.3.i_bh_dr_ig_pl |
| 5 | $C_2 \wr S_5$ | \( 1 + 4 T + 17 T^{2} + 43 T^{3} + 146 T^{4} + 63 p T^{5} + 146 p T^{6} + 43 p^{2} T^{7} + 17 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.5.e_r_br_fq_md |
| 7 | $C_2 \wr S_5$ | \( 1 + 10 T + 62 T^{2} + 271 T^{3} + 960 T^{4} + 2761 T^{5} + 960 p T^{6} + 271 p^{2} T^{7} + 62 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.k_ck_kl_bky_ecf |
| 11 | $C_2 \wr S_5$ | \( 1 + 8 T + 56 T^{2} + 284 T^{3} + 115 p T^{4} + 4371 T^{5} + 115 p^{2} T^{6} + 284 p^{2} T^{7} + 56 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.i_ce_ky_bwr_gmd |
| 13 | $C_2 \wr S_5$ | \( 1 + 8 T + 47 T^{2} + 253 T^{3} + 1176 T^{4} + 4367 T^{5} + 1176 p T^{6} + 253 p^{2} T^{7} + 47 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.i_bv_jt_btg_glz |
| 17 | $C_2 \wr S_5$ | \( 1 - 9 T + 88 T^{2} - 507 T^{3} + 177 p T^{4} - 725 p T^{5} + 177 p^{2} T^{6} - 507 p^{2} T^{7} + 88 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.aj_dk_atn_elt_asgb |
| 19 | $C_2 \wr S_5$ | \( 1 + 16 T + 175 T^{2} + 1297 T^{3} + 7792 T^{4} + 36973 T^{5} + 7792 p T^{6} + 1297 p^{2} T^{7} + 175 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.q_gt_bxx_lns_ccsb |
| 23 | $C_2 \wr S_5$ | \( 1 + T + 88 T^{2} + 110 T^{3} + 3488 T^{4} + 4001 T^{5} + 3488 p T^{6} + 110 p^{2} T^{7} + 88 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.23.b_dk_eg_fee_fxx |
| 29 | $C_2 \wr S_5$ | \( 1 - 2 T + 103 T^{2} - 223 T^{3} + 4774 T^{4} - 9571 T^{5} + 4774 p T^{6} - 223 p^{2} T^{7} + 103 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.ac_dz_aip_hbq_aoed |
| 31 | $C_2 \wr S_5$ | \( 1 - 2 T + 4 p T^{2} - 239 T^{3} + 6976 T^{4} - 10739 T^{5} + 6976 p T^{6} - 239 p^{2} T^{7} + 4 p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.ac_eu_ajf_kii_apxb |
| 37 | $C_2 \wr S_5$ | \( 1 + 17 T + 184 T^{2} + 1241 T^{3} + 7179 T^{4} + 38167 T^{5} + 7179 p T^{6} + 1241 p^{2} T^{7} + 184 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.r_hc_bvt_kqd_celz |
| 41 | $C_2 \wr S_5$ | \( 1 + 5 T + 43 T^{2} - 27 T^{3} + 1157 T^{4} - 2165 T^{5} + 1157 p T^{6} - 27 p^{2} T^{7} + 43 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.f_br_abb_bsn_adfh |
| 43 | $C_2 \wr S_5$ | \( 1 + 26 T + 428 T^{2} + 4964 T^{3} + 45307 T^{4} + 328729 T^{5} + 45307 p T^{6} + 4964 p^{2} T^{7} + 428 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.ba_qm_hiy_cpap_sshl |
| 47 | $C_2 \wr S_5$ | \( 1 - 16 T + 285 T^{2} - 2841 T^{3} + 28646 T^{4} - 196535 T^{5} + 28646 p T^{6} - 2841 p^{2} T^{7} + 285 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.aq_kz_aefh_bqju_aletb |
| 53 | $C_2 \wr S_5$ | \( 1 - 2 T + 126 T^{2} + 59 T^{3} + 7672 T^{4} + 11875 T^{5} + 7672 p T^{6} + 59 p^{2} T^{7} + 126 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.ac_ew_ch_ljc_rot |
| 59 | $C_2 \wr S_5$ | \( 1 + 13 T + 249 T^{2} + 2201 T^{3} + 26025 T^{4} + 176267 T^{5} + 26025 p T^{6} + 2201 p^{2} T^{7} + 249 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.n_jp_dgr_bmmz_katn |
| 61 | $C_2 \wr S_5$ | \( 1 + 3 T + 180 T^{2} + 644 T^{3} + 18018 T^{4} + 51407 T^{5} + 18018 p T^{6} + 644 p^{2} T^{7} + 180 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.d_gy_yu_bara_cybf |
| 67 | $C_2 \wr S_5$ | \( 1 + 40 T + 906 T^{2} + 14060 T^{3} + 165963 T^{4} + 1525665 T^{5} + 165963 p T^{6} + 14060 p^{2} T^{7} + 906 p^{3} T^{8} + 40 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.bo_biw_uuu_jlnf_diuxl |
| 71 | $C_2 \wr S_5$ | \( 1 + 5 T + 234 T^{2} + 382 T^{3} + 22488 T^{4} + 3217 T^{5} + 22488 p T^{6} + 382 p^{2} T^{7} + 234 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.f_ja_os_bhgy_ett |
| 73 | $C_2 \wr S_5$ | \( 1 + 9 T + 216 T^{2} + 936 T^{3} + 18656 T^{4} + 51623 T^{5} + 18656 p T^{6} + 936 p^{2} T^{7} + 216 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.j_ii_bka_bbpo_cyjn |
| 79 | $C_2 \wr S_5$ | \( 1 - 13 T + 232 T^{2} - 1040 T^{3} + 9422 T^{4} + 18513 T^{5} + 9422 p T^{6} - 1040 p^{2} T^{7} + 232 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.an_iy_aboa_nyk_bbkb |
| 83 | $C_2 \wr S_5$ | \( 1 + 17 T + 393 T^{2} + 4501 T^{3} + 61305 T^{4} + 517387 T^{5} + 61305 p T^{6} + 4501 p^{2} T^{7} + 393 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.r_pd_grd_dmrx_bdljn |
| 89 | $C_2 \wr S_5$ | \( 1 - 10 T + 355 T^{2} - 2727 T^{3} + 56332 T^{4} - 336029 T^{5} + 56332 p T^{6} - 2727 p^{2} T^{7} + 355 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.ak_nr_aeax_dfiq_atdcf |
| 97 | $C_2 \wr S_5$ | \( 1 + 42 T + 1138 T^{2} + 20908 T^{3} + 298723 T^{4} + 3282507 T^{5} + 298723 p T^{6} + 20908 p^{2} T^{7} + 1138 p^{3} T^{8} + 42 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.bq_bru_beye_qzxj_hetuh |
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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
−8.308510121795057332287369839122, −8.033973260003417373822184484863, −7.918176970930472177144434517950, −7.82600717664272797900997163559, −7.25291239419469011407395994983, −7.12569140409990635237351569113, −6.85368116353021803487562696802, −6.78508991733640434341388317845, −6.41067259273828530576133322776, −6.30986525984022276018825272093, −5.90920154706839806556755895691, −5.67948241450258179702631112956, −5.66781894712974751554447947100, −5.61645903680078664472003458312, −5.15288105060554207579951834988, −4.85799607648645491448078254509, −4.74853543077398265073303491412, −4.58927791410557317158916104593, −4.26684786120460053621317945923, −3.94750586595729642872362565201, −3.71400212312596032679461739802, −3.22931257657162174863104478197, −2.98265896044668992375611323578, −2.88828168430306557871659675712, −1.81345003103451491646772430105, 0, 0, 0, 0, 0,
1.81345003103451491646772430105, 2.88828168430306557871659675712, 2.98265896044668992375611323578, 3.22931257657162174863104478197, 3.71400212312596032679461739802, 3.94750586595729642872362565201, 4.26684786120460053621317945923, 4.58927791410557317158916104593, 4.74853543077398265073303491412, 4.85799607648645491448078254509, 5.15288105060554207579951834988, 5.61645903680078664472003458312, 5.66781894712974751554447947100, 5.67948241450258179702631112956, 5.90920154706839806556755895691, 6.30986525984022276018825272093, 6.41067259273828530576133322776, 6.78508991733640434341388317845, 6.85368116353021803487562696802, 7.12569140409990635237351569113, 7.25291239419469011407395994983, 7.82600717664272797900997163559, 7.918176970930472177144434517950, 8.033973260003417373822184484863, 8.308510121795057332287369839122
