| L(s) = 1 | + 2-s + 5·3-s − 4-s + 5·5-s + 5·6-s + 2·7-s + 15·9-s + 5·10-s + 5·11-s − 5·12-s + 8·13-s + 2·14-s + 25·15-s + 10·17-s + 15·18-s − 5·20-s + 10·21-s + 5·22-s − 2·23-s + 15·25-s + 8·26-s + 35·27-s − 2·28-s + 7·29-s + 25·30-s + 9·31-s + 25·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.88·3-s − 1/2·4-s + 2.23·5-s + 2.04·6-s + 0.755·7-s + 5·9-s + 1.58·10-s + 1.50·11-s − 1.44·12-s + 2.21·13-s + 0.534·14-s + 6.45·15-s + 2.42·17-s + 3.53·18-s − 1.11·20-s + 2.18·21-s + 1.06·22-s − 0.417·23-s + 3·25-s + 1.56·26-s + 6.73·27-s − 0.377·28-s + 1.29·29-s + 4.56·30-s + 1.61·31-s + 4.35·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(262.5437783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(262.5437783\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{5} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{5} \) | |
| 19 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 - T + p T^{2} - 3 T^{3} + 5 T^{4} - p^{3} T^{5} + 5 p T^{6} - 3 p^{2} T^{7} + p^{4} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.2.ab_c_ad_f_ai |
| 7 | $C_2 \wr S_5$ | \( 1 - 2 T + 12 T^{2} - 20 T^{3} + 79 T^{4} - 60 T^{5} + 79 p T^{6} - 20 p^{2} T^{7} + 12 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.ac_m_au_db_aci |
| 11 | $C_2 \wr S_5$ | \( 1 - 5 T + 23 T^{2} - 72 T^{3} + 410 T^{4} - 1366 T^{5} + 410 p T^{6} - 72 p^{2} T^{7} + 23 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.af_x_acu_pu_acao |
| 13 | $C_2 \wr S_5$ | \( 1 - 8 T + 66 T^{2} - 346 T^{3} + 1685 T^{4} - 6420 T^{5} + 1685 p T^{6} - 346 p^{2} T^{7} + 66 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.ai_co_ani_cmv_ajmy |
| 17 | $C_2 \wr S_5$ | \( 1 - 10 T + 50 T^{2} - 196 T^{3} + 1161 T^{4} - 5908 T^{5} + 1161 p T^{6} - 196 p^{2} T^{7} + 50 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.ak_by_aho_bsr_aitg |
| 23 | $C_2 \wr S_5$ | \( 1 + 2 T + 47 T^{2} + 160 T^{3} + 1270 T^{4} + 5628 T^{5} + 1270 p T^{6} + 160 p^{2} T^{7} + 47 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.c_bv_ge_bww_iim |
| 29 | $C_2 \wr S_5$ | \( 1 - 7 T + 83 T^{2} - 400 T^{3} + 3336 T^{4} - 14434 T^{5} + 3336 p T^{6} - 400 p^{2} T^{7} + 83 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.ah_df_apk_eyi_avje |
| 31 | $C_2 \wr S_5$ | \( 1 - 9 T + 125 T^{2} - 894 T^{3} + 7009 T^{4} - 38247 T^{5} + 7009 p T^{6} - 894 p^{2} T^{7} + 125 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.aj_ev_abik_kjp_acepb |
| 37 | $C_2 \wr S_5$ | \( 1 + 6 T + 98 T^{2} + 416 T^{3} + 5953 T^{4} + 23380 T^{5} + 5953 p T^{6} + 416 p^{2} T^{7} + 98 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.g_du_qa_iuz_bipg |
| 41 | $C_2 \wr S_5$ | \( 1 + 12 T + 229 T^{2} + 1864 T^{3} + 19570 T^{4} + 112696 T^{5} + 19570 p T^{6} + 1864 p^{2} T^{7} + 229 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.m_iv_cts_bcys_gksm |
| 43 | $C_2 \wr S_5$ | \( 1 + 8 T + 216 T^{2} + 1306 T^{3} + 18575 T^{4} + 82860 T^{5} + 18575 p T^{6} + 1306 p^{2} T^{7} + 216 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.i_ii_byg_bbml_esoy |
| 47 | $C_2 \wr S_5$ | \( 1 - 6 T + 82 T^{2} - 410 T^{3} + 4225 T^{4} - 9896 T^{5} + 4225 p T^{6} - 410 p^{2} T^{7} + 82 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.ag_de_apu_ggn_aoqq |
| 53 | $C_2 \wr S_5$ | \( 1 + 8 T + 170 T^{2} + 1086 T^{3} + 13589 T^{4} + 74500 T^{5} + 13589 p T^{6} + 1086 p^{2} T^{7} + 170 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.i_go_bpu_ucr_egfk |
| 59 | $C_2 \wr S_5$ | \( 1 - T + 119 T^{2} + 96 T^{3} + 8426 T^{4} + 22834 T^{5} + 8426 p T^{6} + 96 p^{2} T^{7} + 119 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.59.ab_ep_ds_mmc_bhug |
| 61 | $C_2 \wr S_5$ | \( 1 - 7 T + 132 T^{2} - 521 T^{3} + 9755 T^{4} - 35784 T^{5} + 9755 p T^{6} - 521 p^{2} T^{7} + 132 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.ah_fc_aub_olf_acayi |
| 67 | $C_2 \wr S_5$ | \( 1 - 14 T + 168 T^{2} - 1112 T^{3} + 6163 T^{4} - 31692 T^{5} + 6163 p T^{6} - 1112 p^{2} T^{7} + 168 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.ao_gm_abqu_jdb_abuwy |
| 71 | $C_2 \wr S_5$ | \( 1 - 27 T + 577 T^{2} - 8036 T^{3} + 95560 T^{4} - 862802 T^{5} + 95560 p T^{6} - 8036 p^{2} T^{7} + 577 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.abb_wf_alxc_fljk_abxcis |
| 73 | $C_2 \wr S_5$ | \( 1 - 26 T + 426 T^{2} - 4848 T^{3} + 44229 T^{4} - 384492 T^{5} + 44229 p T^{6} - 4848 p^{2} T^{7} + 426 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.aba_qk_ahem_cnld_avwue |
| 79 | $C_2 \wr S_5$ | \( 1 + 23 T + 354 T^{2} + 2477 T^{3} + 11005 T^{4} - 4968 T^{5} + 11005 p T^{6} + 2477 p^{2} T^{7} + 354 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.x_nq_drh_qhh_ahjc |
| 83 | $C_2 \wr S_5$ | \( 1 + 12 T + 394 T^{2} + 3324 T^{3} + 61357 T^{4} + 384144 T^{5} + 61357 p T^{6} + 3324 p^{2} T^{7} + 394 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.m_pe_exw_dmtx_vwgu |
| 89 | $C_2 \wr S_5$ | \( 1 - 9 T + 313 T^{2} - 2328 T^{3} + 46270 T^{4} - 283038 T^{5} + 46270 p T^{6} - 2328 p^{2} T^{7} + 313 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.aj_mb_adlo_cqlq_aqcsc |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{5} \) | 5.97.k_uf_fwi_ganq_bhadg |
| 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
−4.83241165188445837318853061388, −4.46833573210813958631462871175, −4.36892577373737734072999174520, −4.36656972394844686740856308359, −4.33571019733651494623673640432, −3.77030742875690921237359535823, −3.72095512132286507300597668960, −3.71971767275104789155218137542, −3.59189917781294254172489595591, −3.36435731162544093966857137855, −3.09725519347025551541663215355, −3.04625630065028179613573610342, −2.93823951013180700871385913908, −2.89009359418090848730770209898, −2.24004368354240926938843715732, −2.23576052236733492407079012232, −2.22627011020930451794773391698, −1.81964552155927779464209960618, −1.71343423592962633141955333511, −1.66057061958957648926926640924, −1.46874180672732826845638630457, −1.10305297478285514829812944436, −0.932726770119098533681800958603, −0.882199977243032778929710025783, −0.64859685443543141162276940534,
0.64859685443543141162276940534, 0.882199977243032778929710025783, 0.932726770119098533681800958603, 1.10305297478285514829812944436, 1.46874180672732826845638630457, 1.66057061958957648926926640924, 1.71343423592962633141955333511, 1.81964552155927779464209960618, 2.22627011020930451794773391698, 2.23576052236733492407079012232, 2.24004368354240926938843715732, 2.89009359418090848730770209898, 2.93823951013180700871385913908, 3.04625630065028179613573610342, 3.09725519347025551541663215355, 3.36435731162544093966857137855, 3.59189917781294254172489595591, 3.71971767275104789155218137542, 3.72095512132286507300597668960, 3.77030742875690921237359535823, 4.33571019733651494623673640432, 4.36656972394844686740856308359, 4.36892577373737734072999174520, 4.46833573210813958631462871175, 4.83241165188445837318853061388
