| L(s) = 1 | + 3·3-s + 2·4-s + 2·5-s − 2·7-s + 2·8-s + 3·9-s − 10·11-s + 6·12-s − 7·13-s + 6·15-s + 4·16-s − 17-s + 5·19-s + 4·20-s − 6·21-s + 2·23-s + 6·24-s + 11·25-s − 2·27-s − 4·28-s − 22·29-s + 32·31-s + 6·32-s − 30·33-s − 4·35-s + 6·36-s − 3·37-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s + 0.894·5-s − 0.755·7-s + 0.707·8-s + 9-s − 3.01·11-s + 1.73·12-s − 1.94·13-s + 1.54·15-s + 16-s − 0.242·17-s + 1.14·19-s + 0.894·20-s − 1.30·21-s + 0.417·23-s + 1.22·24-s + 11/5·25-s − 0.384·27-s − 0.755·28-s − 4.08·29-s + 5.74·31-s + 1.06·32-s − 5.22·33-s − 0.676·35-s + 36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.100718333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.100718333\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 - T + T^{2} )^{3} \) | |
| 37 | \( 1 + 3 T - 30 T^{2} - 137 T^{3} - 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | |
| good | 2 | \( 1 - p T^{2} - p T^{3} + p T^{5} + 9 T^{6} + p^{2} T^{7} - p^{4} T^{9} - p^{5} T^{10} + p^{6} T^{12} \) | 6.2.a_ac_ac_a_c_j |
| 5 | \( 1 - 2 T - 7 T^{2} + 4 T^{3} + 8 p T^{4} + 27 T^{5} - 286 T^{6} + 27 p T^{7} + 8 p^{3} T^{8} + 4 p^{3} T^{9} - 7 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.5.ac_ah_e_bo_bb_ala |
| 7 | \( 1 + 2 T - 12 T^{2} - 20 T^{3} + 92 T^{4} + 66 T^{5} - 626 T^{6} + 66 p T^{7} + 92 p^{2} T^{8} - 20 p^{3} T^{9} - 12 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.c_am_au_do_co_ayc |
| 11 | \( ( 1 + 5 T + 26 T^{2} + 112 T^{3} + 26 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.11.k_cz_sq_dnc_oqo_cblo |
| 13 | \( 1 + 7 T + 9 T^{2} + 8 T^{3} + 128 T^{4} - 657 T^{5} - 6191 T^{6} - 657 p T^{7} + 128 p^{2} T^{8} + 8 p^{3} T^{9} + 9 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.h_j_i_ey_azh_ajed |
| 17 | \( 1 + T - 25 T^{2} - 56 T^{3} + 193 T^{4} + 471 T^{5} - 274 T^{6} + 471 p T^{7} + 193 p^{2} T^{8} - 56 p^{3} T^{9} - 25 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | 6.17.b_az_ace_hl_sd_ako |
| 19 | \( 1 - 5 T - 13 T^{2} + 218 T^{3} - 298 T^{4} - 123 p T^{5} + 18873 T^{6} - 123 p^{2} T^{7} - 298 p^{2} T^{8} + 218 p^{3} T^{9} - 13 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.19.af_an_ik_alm_adlx_bbxx |
| 23 | \( ( 1 - T + 50 T^{2} - 14 T^{3} + 50 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.23.ac_dx_aey_hds_ahba_hzzg |
| 29 | \( ( 1 + 11 T + 106 T^{2} + 639 T^{3} + 106 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.29.w_mv_fiw_buni_mpva_cwrhl |
| 31 | \( ( 1 - 16 T + 157 T^{2} - 1000 T^{3} + 157 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.31.abg_vy_akke_duff_abcmjw_gtfzw |
| 41 | \( 1 - 87 T^{2} - 54 T^{3} + 4002 T^{4} + 2349 T^{5} - 171758 T^{6} + 2349 p T^{7} + 4002 p^{2} T^{8} - 54 p^{3} T^{9} - 87 p^{4} T^{10} + p^{6} T^{12} \) | 6.41.a_adj_acc_fxy_dmj_ajucc |
| 43 | \( ( 1 + 3 T + 50 T^{2} + 284 T^{3} + 50 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.43.g_ef_bhk_mpc_czne_bbtvk |
| 47 | \( ( 1 + 4 T - 6 T^{2} - 138 T^{3} - 6 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.47.i_e_amm_acku_zgq_rcxu |
| 53 | \( 1 - 6 T - 74 T^{2} + 320 T^{3} + 3330 T^{4} - 794 T^{5} - 204570 T^{6} - 794 p T^{7} + 3330 p^{2} T^{8} + 320 p^{3} T^{9} - 74 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.ag_acw_mi_eyc_abeo_alqqc |
| 59 | \( 1 + 3 T - 33 T^{2} + 930 T^{3} + 492 T^{4} - 21309 T^{5} + 436957 T^{6} - 21309 p T^{7} + 492 p^{2} T^{8} + 930 p^{3} T^{9} - 33 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.d_abh_bju_sy_abfnp_ywkb |
| 61 | \( 1 - 9 T - 72 T^{2} + 7 p T^{3} + 5958 T^{4} - 3501 T^{5} - 461784 T^{6} - 3501 p T^{7} + 5958 p^{2} T^{8} + 7 p^{4} T^{9} - 72 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.aj_acu_ql_ive_afer_abahcy |
| 67 | \( 1 + 17 T + p T^{2} + 182 T^{3} + 3878 T^{4} - 30691 T^{5} - 772207 T^{6} - 30691 p T^{7} + 3878 p^{2} T^{8} + 182 p^{3} T^{9} + p^{5} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.r_cp_ha_fte_abtkl_abryih |
| 71 | \( 1 - 18 T + 67 T^{2} + 22 T^{3} + 5970 T^{4} - 31114 T^{5} - 205965 T^{6} - 31114 p T^{7} + 5970 p^{2} T^{8} + 22 p^{3} T^{9} + 67 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.as_cp_w_ivq_abuas_alsrt |
| 73 | \( ( 1 - 6 T + 167 T^{2} - 692 T^{3} + 167 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.73.am_og_afai_dlpv_azcxs_mnagy |
| 79 | \( 1 - 3 T - 195 T^{2} + 212 T^{3} + 24264 T^{4} - 6075 T^{5} - 2188011 T^{6} - 6075 p T^{7} + 24264 p^{2} T^{8} + 212 p^{3} T^{9} - 195 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.ad_ahn_ie_bjxg_aizr_aeumsh |
| 83 | \( 1 - 12 T - 89 T^{2} + 932 T^{3} + 12582 T^{4} - 52844 T^{5} - 879933 T^{6} - 52844 p T^{7} + 12582 p^{2} T^{8} + 932 p^{3} T^{9} - 89 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.am_adl_bjw_spy_adaem_abybrp |
| 89 | \( 1 + 5 T - 128 T^{2} - 1647 T^{3} + 3762 T^{4} + 87953 T^{5} + 582948 T^{6} + 87953 p T^{7} + 3762 p^{2} T^{8} - 1647 p^{3} T^{9} - 128 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.f_aey_aclj_fos_facv_bhejc |
| 97 | \( ( 1 - 10 T + 157 T^{2} - 799 T^{3} + 157 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.97.au_py_ahag_ebed_abqibu_tzidx |
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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
−7.69993610041469357266979247047, −7.53406344413462939512271602355, −7.45617617526616811428065223340, −7.01281988223135416033084709051, −7.00963944199211142845455508519, −6.69793777359335802681295178485, −6.63593028287627992614781074719, −6.41947554631521485018158568879, −5.83035143662122257329875767734, −5.72966168144407160104184375710, −5.46506210962650879112164859909, −5.32660256751921637086109576151, −5.06419399090337770344450430151, −4.87590553147170976924388543329, −4.78781104098359710563007834515, −4.15402561676127557261596016549, −3.97797794782956068920900802576, −3.65755768126112717559563042102, −3.11375430731684129566409455960, −2.77526622111301191789301233034, −2.71909913962644326315209349837, −2.67813993635832034016648407591, −2.49712916652037064508259919535, −1.96173313570985554853488226376, −1.32255602229405317333060190253,
1.32255602229405317333060190253, 1.96173313570985554853488226376, 2.49712916652037064508259919535, 2.67813993635832034016648407591, 2.71909913962644326315209349837, 2.77526622111301191789301233034, 3.11375430731684129566409455960, 3.65755768126112717559563042102, 3.97797794782956068920900802576, 4.15402561676127557261596016549, 4.78781104098359710563007834515, 4.87590553147170976924388543329, 5.06419399090337770344450430151, 5.32660256751921637086109576151, 5.46506210962650879112164859909, 5.72966168144407160104184375710, 5.83035143662122257329875767734, 6.41947554631521485018158568879, 6.63593028287627992614781074719, 6.69793777359335802681295178485, 7.00963944199211142845455508519, 7.01281988223135416033084709051, 7.45617617526616811428065223340, 7.53406344413462939512271602355, 7.69993610041469357266979247047
Plot not available for L-functions of degree greater than 10.
