| L(s) = 1 | − 3·9-s − 6·11-s − 6·19-s − 18·29-s + 18·31-s + 24·49-s − 48·59-s − 18·61-s + 7·64-s + 12·71-s − 30·79-s + 6·81-s + 6·89-s + 18·99-s + 12·101-s − 27·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + 18·171-s + ⋯ |
| L(s) = 1 | − 9-s − 1.80·11-s − 1.37·19-s − 3.34·29-s + 3.23·31-s + 24/7·49-s − 6.24·59-s − 2.30·61-s + 7/8·64-s + 1.42·71-s − 3.37·79-s + 2/3·81-s + 0.635·89-s + 1.80·99-s + 1.19·101-s − 2.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.38·169-s + 1.37·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2115555576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2115555576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) | |
| 5 | \( 1 \) | |
| 19 | \( ( 1 + T )^{6} \) | |
| good | 2 | \( 1 - 7 T^{6} + p^{6} T^{12} \) | 6.2.a_a_a_a_a_ah |
| 7 | \( 1 - 24 T^{2} + 312 T^{4} - 2686 T^{6} + 312 p^{2} T^{8} - 24 p^{4} T^{10} + p^{6} T^{12} \) | 6.7.a_ay_a_ma_a_adzi |
| 11 | \( ( 1 + 3 T + 27 T^{2} + 54 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.11.g_cl_kk_clj_iaq_bjfe |
| 13 | \( 1 - 18 T^{2} + 327 T^{4} - 2716 T^{6} + 327 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) | 6.13.a_as_a_mp_a_aeam |
| 17 | \( ( 1 - p T^{2} )^{6} \) | 6.17.a_ady_a_gkt_a_afpjg |
| 23 | \( 1 - 81 T^{2} + 3195 T^{4} - 85462 T^{6} + 3195 p^{2} T^{8} - 81 p^{4} T^{10} + p^{6} T^{12} \) | 6.23.a_add_a_esx_a_aewla |
| 29 | \( ( 1 + 3 T + p T^{2} )^{6} \) | 6.29.s_lx_ere_brqg_ljpi_csopp |
| 31 | \( ( 1 - 9 T + 3 p T^{2} - 542 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.31.as_kh_aecc_bjtr_ajrmq_chmgo |
| 37 | \( 1 - 78 T^{2} + 3639 T^{4} - 123172 T^{6} + 3639 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) | 6.37.a_ada_a_fjz_a_ahafk |
| 41 | \( ( 1 + 426 T^{3} + p^{3} T^{6} )^{2} \) | 6.41.a_a_bgu_a_a_sejm |
| 43 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{3} \) | 6.43.a_aco_a_kjf_a_aomuu |
| 47 | \( 1 - 126 T^{2} + 6255 T^{4} - 229732 T^{6} + 6255 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) | 6.47.a_aew_a_jgp_a_anbvw |
| 53 | \( 1 - 99 T^{2} + 1710 T^{4} + 81929 T^{6} + 1710 p^{2} T^{8} - 99 p^{4} T^{10} + p^{6} T^{12} \) | 6.53.a_adv_a_cnu_a_erfd |
| 59 | \( ( 1 + 24 T + 360 T^{2} + 3276 T^{3} + 360 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.59.bw_bxw_bjgq_stem_htsou_coqggc |
| 61 | \( ( 1 + 9 T + 174 T^{2} + 985 T^{3} + 174 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.61.s_qn_hog_dykw_bieuc_mnmnl |
| 67 | \( 1 - 189 T^{2} + 12951 T^{4} - 631654 T^{6} + 12951 p^{2} T^{8} - 189 p^{4} T^{10} + p^{6} T^{12} \) | 6.67.a_ahh_a_ted_a_abjykk |
| 71 | \( ( 1 - 6 T + 132 T^{2} - 330 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.71.am_lo_adii_chjk_aousy_iaizq |
| 73 | \( 1 - 243 T^{2} + 31062 T^{4} - 2713783 T^{6} + 31062 p^{2} T^{8} - 243 p^{4} T^{10} + p^{6} T^{12} \) | 6.73.a_ajj_a_btys_a_afykmh |
| 79 | \( ( 1 + 15 T + 189 T^{2} + 1454 T^{3} + 189 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.79.be_xf_mry_gfod_cpkqe_zhhpi |
| 83 | \( 1 - 405 T^{2} + 74295 T^{4} - 7889398 T^{6} + 74295 p^{2} T^{8} - 405 p^{4} T^{10} + p^{6} T^{12} \) | 6.83.a_app_a_efxn_a_argwsk |
| 89 | \( ( 1 - 3 T + 234 T^{2} - 531 T^{3} + 234 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.89.ag_sj_adqw_frio_axyty_zixsv |
| 97 | \( 1 - 102 T^{2} + 17871 T^{4} - 1276180 T^{6} + 17871 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) | 6.97.a_ady_a_balj_a_acupvw |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.99269128213850589663704260327, −4.89973613435867959850572977958, −4.75663136546294923284337434527, −4.50501872132703961025981465027, −4.31340984859887761150705542571, −4.25727924540442761406113375008, −4.24840154752848861300659770185, −3.99270292245372902438077201640, −3.62627662543527794935278722014, −3.51478549290231443997373577788, −3.22879120105861263629041757213, −3.22504461384550954314554921167, −2.99430054008238813575431285305, −2.95244250507849088387809062583, −2.57986025141102418082025846706, −2.56050305336771329513769182876, −2.21604732456567211189139997190, −2.12650462002888581291474442195, −2.05553751563134065071683262755, −1.60738563326235157121508974625, −1.50646178088522376899480164672, −1.17900201547043981912504346540, −0.833865326968442660366497955383, −0.41212784880856727998113936686, −0.097156874649593538007530483222,
0.097156874649593538007530483222, 0.41212784880856727998113936686, 0.833865326968442660366497955383, 1.17900201547043981912504346540, 1.50646178088522376899480164672, 1.60738563326235157121508974625, 2.05553751563134065071683262755, 2.12650462002888581291474442195, 2.21604732456567211189139997190, 2.56050305336771329513769182876, 2.57986025141102418082025846706, 2.95244250507849088387809062583, 2.99430054008238813575431285305, 3.22504461384550954314554921167, 3.22879120105861263629041757213, 3.51478549290231443997373577788, 3.62627662543527794935278722014, 3.99270292245372902438077201640, 4.24840154752848861300659770185, 4.25727924540442761406113375008, 4.31340984859887761150705542571, 4.50501872132703961025981465027, 4.75663136546294923284337434527, 4.89973613435867959850572977958, 4.99269128213850589663704260327
Plot not available for L-functions of degree greater than 10.
