| L(s) = 1 | − 3·5-s − 6·8-s − 3·13-s + 12·17-s + 30·19-s + 3·23-s + 6·25-s − 3·31-s − 6·37-s + 18·40-s + 3·41-s − 6·43-s − 6·47-s + 9·49-s − 18·53-s + 9·59-s + 12·61-s + 11·64-s + 9·65-s − 15·67-s − 6·71-s + 24·73-s + 6·79-s + 3·83-s − 36·85-s + 6·89-s − 90·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 2.12·8-s − 0.832·13-s + 2.91·17-s + 6.88·19-s + 0.625·23-s + 6/5·25-s − 0.538·31-s − 0.986·37-s + 2.84·40-s + 0.468·41-s − 0.914·43-s − 0.875·47-s + 9/7·49-s − 2.47·53-s + 1.17·59-s + 1.53·61-s + 11/8·64-s + 1.11·65-s − 1.83·67-s − 0.712·71-s + 2.80·73-s + 0.675·79-s + 0.329·83-s − 3.90·85-s + 0.635·89-s − 9.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.085647831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.085647831\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( ( 1 + T + T^{2} )^{3} \) | |
| good | 2 | \( ( 1 + 3 T^{3} + p^{3} T^{6} )^{2} \) | 6.2.a_a_g_a_a_z |
| 5 | \( 1 + 3 T + 3 T^{2} - 6 T^{3} - 6 p T^{4} - 33 T^{5} - 11 T^{6} - 33 p T^{7} - 6 p^{3} T^{8} - 6 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.5.d_d_ag_abe_abh_al |
| 7 | \( 1 - 9 T^{2} + 2 p T^{3} + 18 T^{4} - 9 p T^{5} + 24 p T^{6} - 9 p^{2} T^{7} + 18 p^{2} T^{8} + 2 p^{4} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} \) | 6.7.a_aj_o_s_acl_gm |
| 11 | \( 1 - 27 T^{2} + 6 T^{3} + 432 T^{4} - 81 T^{5} - 5348 T^{6} - 81 p T^{7} + 432 p^{2} T^{8} + 6 p^{3} T^{9} - 27 p^{4} T^{10} + p^{6} T^{12} \) | 6.11.a_abb_g_qq_add_ahxs |
| 17 | \( ( 1 - 6 T + 54 T^{2} - 198 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.17.am_fo_aboe_koe_acbbo_jpzi |
| 19 | \( ( 1 - 5 T + p T^{2} )^{6} \) | 6.19.abe_sv_ahxu_cmba_apiti_cxepl |
| 23 | \( 1 - 3 T - 27 T^{2} + 264 T^{3} - 90 T^{4} - 3315 T^{5} + 24361 T^{6} - 3315 p T^{7} - 90 p^{2} T^{8} + 264 p^{3} T^{9} - 27 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.ad_abb_ke_adm_aexn_bkaz |
| 29 | \( 1 - 81 T^{2} + 6 T^{3} + 4212 T^{4} - 243 T^{5} - 141482 T^{6} - 243 p T^{7} + 4212 p^{2} T^{8} + 6 p^{3} T^{9} - 81 p^{4} T^{10} + p^{6} T^{12} \) | 6.29.a_add_g_gga_ajj_aibhq |
| 31 | \( 1 + 3 T - 75 T^{2} - 112 T^{3} + 126 p T^{4} + 2547 T^{5} - 134679 T^{6} + 2547 p T^{7} + 126 p^{3} T^{8} - 112 p^{3} T^{9} - 75 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.d_acx_aei_fug_dtz_ahrfz |
| 37 | \( ( 1 + 3 T + 60 T^{2} + 250 T^{3} + 60 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.37.g_ez_bhc_ocy_cygc_zwwe |
| 41 | \( 1 - 3 T - 99 T^{2} + 192 T^{3} + 6462 T^{4} - 6339 T^{5} - 288947 T^{6} - 6339 p T^{7} + 6462 p^{2} T^{8} + 192 p^{3} T^{9} - 99 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.ad_adv_hk_joo_ajjv_aqllj |
| 43 | \( 1 + 6 T - 24 T^{2} - 580 T^{3} - 1800 T^{4} + 8334 T^{5} + 158550 T^{6} + 8334 p T^{7} - 1800 p^{2} T^{8} - 580 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.g_ay_awi_acrg_mio_jaoc |
| 47 | \( 1 + 6 T - 78 T^{2} - 156 T^{3} + 5118 T^{4} - 3162 T^{5} - 306290 T^{6} - 3162 p T^{7} + 5118 p^{2} T^{8} - 156 p^{3} T^{9} - 78 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.g_ada_aga_how_aerq_arlck |
| 53 | \( ( 1 + 9 T + 105 T^{2} + 846 T^{3} + 105 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.53.s_lf_fhu_cdhv_srum_fubdy |
| 59 | \( 1 - 9 T - 27 T^{2} + 246 T^{3} - 189 T^{4} + 17595 T^{5} - 188498 T^{6} + 17595 p T^{7} - 189 p^{2} T^{8} + 246 p^{3} T^{9} - 27 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.aj_abb_jm_ahh_baat_aksvy |
| 61 | \( 1 - 12 T + 75 T^{2} - 634 T^{3} - 612 T^{4} + 49671 T^{5} - 362874 T^{6} + 49671 p T^{7} - 612 p^{2} T^{8} - 634 p^{3} T^{9} + 75 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.am_cx_ayk_axo_cvml_auqus |
| 67 | \( 1 + 15 T + 111 T^{2} + 626 T^{3} - 2772 T^{4} - 100017 T^{5} - 985467 T^{6} - 100017 p T^{7} - 2772 p^{2} T^{8} + 626 p^{3} T^{9} + 111 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.p_eh_yc_aecq_afryv_acebup |
| 71 | \( ( 1 + 3 T + 180 T^{2} + 423 T^{3} + 180 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.71.g_of_cwc_dlmw_otlm_mfvth |
| 73 | \( ( 1 - 12 T + 174 T^{2} - 1100 T^{3} + 174 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.73.ay_sy_ajlg_erky_abuklc_rjyde |
| 79 | \( 1 - 6 T - 120 T^{2} + 1292 T^{3} + 5328 T^{4} - 63342 T^{5} + 79854 T^{6} - 63342 p T^{7} + 5328 p^{2} T^{8} + 1292 p^{3} T^{9} - 120 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.ag_aeq_bxs_hwy_adpsg_eodi |
| 83 | \( 1 - 3 T - 51 T^{2} + 1698 T^{3} - 4197 T^{4} - 47127 T^{5} + 1664494 T^{6} - 47127 p T^{7} - 4197 p^{2} T^{8} + 1698 p^{3} T^{9} - 51 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.ad_abz_cni_agfl_acrsp_dqsha |
| 89 | \( ( 1 - 3 T + 198 T^{2} - 396 T^{3} + 198 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.89.ag_pp_acye_ejqu_arqry_tahoe |
| 97 | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{3} \) | 6.97.y_dp_bwi_doig_baxwq_bbpkb |
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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
−5.27921893716487372106963088601, −5.22749487491414879726599543058, −4.99197252234261220773604988183, −4.98118049083401903337190225115, −4.77572121024437055220554943908, −4.38552134237343362123468940930, −4.23606175558548633571128903907, −3.89194398351291227978631293242, −3.69277462710480871814779324823, −3.68807058304094625352310998587, −3.56917396386046674274723323423, −3.19392250417557257747809112698, −3.07450550472479072319913934667, −3.06673672800938655112565369616, −3.04459549259478806367463852030, −2.98818375184754345864478066841, −2.69378938211002563038456487005, −2.22797598443752637073690231221, −1.97317253726885696087539459021, −1.56447527131768840832279742159, −1.49722730034174363330210236441, −1.08185472752839115898332689024, −0.801497261331528440698142395493, −0.77804341311710632258455284161, −0.41214241452723133882945457881,
0.41214241452723133882945457881, 0.77804341311710632258455284161, 0.801497261331528440698142395493, 1.08185472752839115898332689024, 1.49722730034174363330210236441, 1.56447527131768840832279742159, 1.97317253726885696087539459021, 2.22797598443752637073690231221, 2.69378938211002563038456487005, 2.98818375184754345864478066841, 3.04459549259478806367463852030, 3.06673672800938655112565369616, 3.07450550472479072319913934667, 3.19392250417557257747809112698, 3.56917396386046674274723323423, 3.68807058304094625352310998587, 3.69277462710480871814779324823, 3.89194398351291227978631293242, 4.23606175558548633571128903907, 4.38552134237343362123468940930, 4.77572121024437055220554943908, 4.98118049083401903337190225115, 4.99197252234261220773604988183, 5.22749487491414879726599543058, 5.27921893716487372106963088601
Plot not available for L-functions of degree greater than 10.
