| L(s) = 1 | − 5·4-s + 4·5-s − 10·7-s + 8-s + 8·11-s − 8·13-s + 12·16-s − 9·17-s − 16·19-s − 20·20-s + 23-s − 25-s + 50·28-s − 2·29-s + 2·31-s − 7·32-s − 40·35-s − 17·37-s + 4·40-s + 5·41-s − 26·43-s − 40·44-s − 16·47-s + 38·49-s + 40·52-s − 2·53-s + 32·55-s + ⋯ |
| L(s) = 1 | − 5/2·4-s + 1.78·5-s − 3.77·7-s + 0.353·8-s + 2.41·11-s − 2.21·13-s + 3·16-s − 2.18·17-s − 3.67·19-s − 4.47·20-s + 0.208·23-s − 1/5·25-s + 9.44·28-s − 0.371·29-s + 0.359·31-s − 1.23·32-s − 6.76·35-s − 2.79·37-s + 0.632·40-s + 0.780·41-s − 3.96·43-s − 6.03·44-s − 2.33·47-s + 38/7·49-s + 5.54·52-s − 0.274·53-s + 4.31·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 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_ab_n_ad |
| 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.ae_r_abr_fq_amd |
| 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.ai_ce_aky_bwr_agmd |
| 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.j_dk_tn_elt_sgb |
| 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.ab_dk_aeg_fee_afxx |
| 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.c_dz_ip_hbq_oed |
| 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.af_br_bb_bsn_dfh |
| 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.q_kz_efh_bqju_letb |
| 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.c_ew_ach_ljc_arot |
| 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.an_jp_adgr_bmmz_akatn |
| 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.af_ja_aos_bhgy_aett |
| 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.ar_pd_agrd_dmrx_abdljn |
| 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.k_nr_eax_dfiq_tdcf |
| 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
−6.11263098951157281649375965800, −5.99503083200016249065506950790, −5.65227587416626159609273593857, −5.37771439134333972435456779884, −5.34992750411783394736430205955, −5.05458579281849048624227065689, −4.98068055432181895172756634071, −4.60207279226395876975706763406, −4.57497754629458484831789095765, −4.56307526098948429281309033678, −4.10458390144791444496311350686, −4.02355533156373497569480676469, −3.91996002646575048528901937851, −3.80861587133731933471901943036, −3.51477236892364351739860016826, −3.20831411199908607572535634107, −3.09590320782381055363700619206, −3.05594592787848585099927383430, −2.50573680534212746696530802656, −2.34840975633288460840948023835, −2.25579789890075286093777542112, −1.79492876172999938679839527386, −1.78514281120747373697537744558, −1.44197007533585499351705822511, −1.28218494958583491295565182889, 0, 0, 0, 0, 0,
1.28218494958583491295565182889, 1.44197007533585499351705822511, 1.78514281120747373697537744558, 1.79492876172999938679839527386, 2.25579789890075286093777542112, 2.34840975633288460840948023835, 2.50573680534212746696530802656, 3.05594592787848585099927383430, 3.09590320782381055363700619206, 3.20831411199908607572535634107, 3.51477236892364351739860016826, 3.80861587133731933471901943036, 3.91996002646575048528901937851, 4.02355533156373497569480676469, 4.10458390144791444496311350686, 4.56307526098948429281309033678, 4.57497754629458484831789095765, 4.60207279226395876975706763406, 4.98068055432181895172756634071, 5.05458579281849048624227065689, 5.34992750411783394736430205955, 5.37771439134333972435456779884, 5.65227587416626159609273593857, 5.99503083200016249065506950790, 6.11263098951157281649375965800
