| L(s) = 1 | + 2·4-s + 6·5-s + 5·7-s − 4·8-s + 4·11-s + 3·13-s + 4·16-s − 4·17-s + 12·20-s + 8·23-s + 21·25-s + 10·28-s + 6·29-s − 18·31-s − 12·32-s + 30·35-s + 14·37-s − 24·40-s − 20·41-s + 15·43-s + 8·44-s + 8·49-s + 6·52-s + 32·53-s + 24·55-s − 20·56-s + 10·59-s + ⋯ |
| L(s) = 1 | + 4-s + 2.68·5-s + 1.88·7-s − 1.41·8-s + 1.20·11-s + 0.832·13-s + 16-s − 0.970·17-s + 2.68·20-s + 1.66·23-s + 21/5·25-s + 1.88·28-s + 1.11·29-s − 3.23·31-s − 2.12·32-s + 5.07·35-s + 2.30·37-s − 3.79·40-s − 3.12·41-s + 2.28·43-s + 1.20·44-s + 8/7·49-s + 0.832·52-s + 4.39·53-s + 3.23·55-s − 2.67·56-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.42496582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.42496582\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( ( 1 - T )^{6} \) | |
| 13 | \( 1 - 3 T + 12 T^{2} - 47 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | |
| good | 2 | \( 1 - p T^{2} + p^{2} T^{3} - p^{2} T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{5} T^{9} - p^{5} T^{10} + p^{6} T^{12} \) | 6.2.a_ac_e_a_ae_m |
| 7 | \( 1 - 5 T + 17 T^{2} - 34 T^{3} - p T^{4} + 291 T^{5} - 978 T^{6} + 291 p T^{7} - p^{3} T^{8} - 34 p^{3} T^{9} + 17 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.af_r_abi_ah_lf_ablq |
| 11 | \( 1 - 4 T - T^{2} + 4 p T^{3} - 122 T^{4} + 60 T^{5} + 683 T^{6} + 60 p T^{7} - 122 p^{2} T^{8} + 4 p^{4} T^{9} - p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.ae_ab_bs_aes_ci_bah |
| 17 | \( 1 + 4 T - 27 T^{2} - 32 T^{3} + 710 T^{4} - 18 p T^{5} - 931 p T^{6} - 18 p^{2} T^{7} + 710 p^{2} T^{8} - 32 p^{3} T^{9} - 27 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.e_abb_abg_bbi_alu_axkt |
| 19 | \( 1 - 17 T^{2} + 8 p T^{3} - 34 T^{4} - 68 p T^{5} + 737 p T^{6} - 68 p^{2} T^{7} - 34 p^{2} T^{8} + 8 p^{4} T^{9} - 17 p^{4} T^{10} + p^{6} T^{12} \) | 6.19.a_ar_fw_abi_abxs_usp |
| 23 | \( 1 - 8 T + 35 T^{2} - 104 T^{3} - 338 T^{4} + 4584 T^{5} - 24769 T^{6} + 4584 p T^{7} - 338 p^{2} T^{8} - 104 p^{3} T^{9} + 35 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.ai_bj_aea_ana_gui_abkqr |
| 29 | \( 1 - 6 T - 33 T^{2} + 6 p T^{3} + 996 T^{4} - 1320 T^{5} - 33131 T^{6} - 1320 p T^{7} + 996 p^{2} T^{8} + 6 p^{4} T^{9} - 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.ag_abh_gs_bmi_abyu_abxah |
| 31 | \( ( 1 + 9 T + 92 T^{2} + 449 T^{3} + 92 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.31.s_kf_dug_bgxu_ipti_cboyr |
| 37 | \( 1 - 14 T + 73 T^{2} + 46 T^{3} - 1798 T^{4} - 470 T^{5} + 54665 T^{6} - 470 p T^{7} - 1798 p^{2} T^{8} + 46 p^{3} T^{9} + 73 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.ao_cv_bu_acre_asc_dcwn |
| 41 | \( 1 + 20 T + 159 T^{2} + 1112 T^{3} + 11120 T^{4} + 85194 T^{5} + 521437 T^{6} + 85194 p T^{7} + 11120 p^{2} T^{8} + 1112 p^{3} T^{9} + 159 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.u_gd_bqu_qls_ewas_bdrjh |
| 43 | \( 1 - 15 T + 55 T^{2} - 184 T^{3} + 3845 T^{4} - 9545 T^{5} - 100222 T^{6} - 9545 p T^{7} + 3845 p^{2} T^{8} - 184 p^{3} T^{9} + 55 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ap_cd_ahc_frx_aodd_afsgs |
| 47 | \( ( 1 + 137 T^{2} + 2 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) | 6.47.a_ko_e_buvf_vc_eifbs |
| 53 | \( ( 1 - 16 T + 191 T^{2} - 1712 T^{3} + 191 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.53.abg_yo_aocu_giyx_acitpo_srocq |
| 59 | \( 1 - 10 T + 25 T^{2} - 330 T^{3} - 900 T^{4} + 37940 T^{5} - 225117 T^{6} + 37940 p T^{7} - 900 p^{2} T^{8} - 330 p^{3} T^{9} + 25 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.ak_z_ams_abiq_cedg_amvaj |
| 61 | \( 1 + 9 T - 101 T^{2} - 340 T^{3} + 13121 T^{4} + 13651 T^{5} - 889882 T^{6} + 13651 p T^{7} + 13121 p^{2} T^{8} - 340 p^{3} T^{9} - 101 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.j_adx_anc_tkr_ufb_abyqkg |
| 67 | \( 1 - 11 T - 37 T^{2} + 716 T^{3} + 881 T^{4} - 10353 T^{5} - 108702 T^{6} - 10353 p T^{7} + 881 p^{2} T^{8} + 716 p^{3} T^{9} - 37 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.al_abl_bbo_bhx_apif_ageuw |
| 71 | \( 1 - 4 T - 179 T^{2} + 264 T^{3} + 22092 T^{4} - 11194 T^{5} - 1792125 T^{6} - 11194 p T^{7} + 22092 p^{2} T^{8} + 264 p^{3} T^{9} - 179 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.ae_agx_ke_bgrs_aqoo_adxzbx |
| 73 | \( ( 1 + 15 T + 290 T^{2} + 2293 T^{3} + 290 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.73.be_bez_tra_lcuw_eqxji_btilvx |
| 79 | \( ( 1 - 17 T + 300 T^{2} - 2753 T^{3} + 300 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.79.abi_bif_axgc_ndss_afvxkk_cftpzt |
| 83 | \( ( 1 + 2 T + 237 T^{2} + 324 T^{3} + 237 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.83.e_sk_cjk_fnfj_ouiy_xgrmi |
| 89 | \( 1 + 22 T + 131 T^{2} + 202 T^{3} + 9988 T^{4} + 126960 T^{5} + 789497 T^{6} + 126960 p T^{7} + 9988 p^{2} T^{8} + 202 p^{3} T^{9} + 131 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.w_fb_hu_oue_hfvc_bsxxh |
| 97 | \( 1 - T - 139 T^{2} + 1342 T^{3} + 5279 T^{4} - 85989 T^{5} + 463818 T^{6} - 85989 p T^{7} + 5279 p^{2} T^{8} + 1342 p^{3} T^{9} - 139 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) | 6.97.ab_afj_bzq_hvb_aexfh_bakde |
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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
−5.70663071490076141156907856978, −5.59261807413824364623739145917, −5.47027744586813705023026107932, −5.46904282915010910932786799375, −5.19752857962159650270072640070, −4.91721228583677216610249372636, −4.69026721571852999025120123507, −4.68903763139646480866017061516, −4.44810005588498554361470448926, −4.07312478598554639064216369260, −3.91324664431855237809941688716, −3.61571519673770072679774447704, −3.55873898174486966155593737504, −3.50212815679732663165076297759, −2.99320014475872109041001728101, −2.79674985643431996045847117404, −2.56499271330492702621251030318, −2.46152482768045292567659913459, −2.20621178405810383008677485513, −2.08706606033668870900517371006, −1.72323436045480070892404263538, −1.48585799905733150859954417691, −1.42873053028325911166119081858, −0.936018703175788436475580126518, −0.77187670609570541346524471998,
0.77187670609570541346524471998, 0.936018703175788436475580126518, 1.42873053028325911166119081858, 1.48585799905733150859954417691, 1.72323436045480070892404263538, 2.08706606033668870900517371006, 2.20621178405810383008677485513, 2.46152482768045292567659913459, 2.56499271330492702621251030318, 2.79674985643431996045847117404, 2.99320014475872109041001728101, 3.50212815679732663165076297759, 3.55873898174486966155593737504, 3.61571519673770072679774447704, 3.91324664431855237809941688716, 4.07312478598554639064216369260, 4.44810005588498554361470448926, 4.68903763139646480866017061516, 4.69026721571852999025120123507, 4.91721228583677216610249372636, 5.19752857962159650270072640070, 5.46904282915010910932786799375, 5.47027744586813705023026107932, 5.59261807413824364623739145917, 5.70663071490076141156907856978
Plot not available for L-functions of degree greater than 10.
