| L(s) = 1 | − 3·2-s + 4-s − 3·5-s − 7-s + 7·8-s + 9·10-s − 6·11-s − 3·13-s + 3·14-s − 12·16-s − 5·17-s + 19-s − 3·20-s + 18·22-s + 6·23-s − 9·25-s + 9·26-s − 28-s − 6·29-s − 8·31-s + 7·32-s + 15·34-s + 3·35-s + 2·37-s − 3·38-s − 21·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s + 2.47·8-s + 2.84·10-s − 1.80·11-s − 0.832·13-s + 0.801·14-s − 3·16-s − 1.21·17-s + 0.229·19-s − 0.670·20-s + 3.83·22-s + 1.25·23-s − 9/5·25-s + 1.76·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 1.23·32-s + 2.57·34-s + 0.507·35-s + 0.328·37-s − 0.486·38-s − 3.32·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( ( 1 + T )^{6} \) | |
| 23 | \( ( 1 - T )^{6} \) | |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} + 7 p T^{3} + 25 T^{4} + 17 p T^{5} + 53 T^{6} + 17 p^{2} T^{7} + 25 p^{2} T^{8} + 7 p^{4} T^{9} + p^{7} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.2.d_i_o_z_bi_cb |
| 5 | \( 1 + 3 T + 18 T^{2} + 2 p^{2} T^{3} + 173 T^{4} + 83 p T^{5} + 1048 T^{6} + 83 p^{2} T^{7} + 173 p^{2} T^{8} + 2 p^{5} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.5.d_s_by_gr_pz_boi |
| 7 | \( 1 + T + 24 T^{2} + 6 p T^{3} + 43 p T^{4} + 545 T^{5} + 2580 T^{6} + 545 p T^{7} + 43 p^{3} T^{8} + 6 p^{4} T^{9} + 24 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | 6.7.b_y_bq_lp_uz_dvg |
| 13 | \( 1 + 3 T + 45 T^{2} + 101 T^{3} + 1045 T^{4} + 2187 T^{5} + 16856 T^{6} + 2187 p T^{7} + 1045 p^{2} T^{8} + 101 p^{3} T^{9} + 45 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.d_bt_dx_bof_dgd_yyi |
| 17 | \( 1 + 5 T + 54 T^{2} + 172 T^{3} + 973 T^{4} + 2143 T^{5} + 11992 T^{6} + 2143 p T^{7} + 973 p^{2} T^{8} + 172 p^{3} T^{9} + 54 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.f_cc_gq_bll_del_rtg |
| 19 | \( 1 - T + 13 T^{2} + 58 T^{3} + 451 T^{4} + 49 p T^{5} + 7854 T^{6} + 49 p^{2} T^{7} + 451 p^{2} T^{8} + 58 p^{3} T^{9} + 13 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) | 6.19.ab_n_cg_rj_bjv_lqc |
| 29 | \( 1 + 6 T + 66 T^{2} + 279 T^{3} + 2908 T^{4} + 12884 T^{5} + 100128 T^{6} + 12884 p T^{7} + 2908 p^{2} T^{8} + 279 p^{3} T^{9} + 66 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.g_co_kt_ehw_tbo_fsdc |
| 31 | \( 1 + 8 T + 104 T^{2} + 665 T^{3} + 5648 T^{4} + 33680 T^{5} + 219486 T^{6} + 33680 p T^{7} + 5648 p^{2} T^{8} + 665 p^{3} T^{9} + 104 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.i_ea_zp_ijg_bxvk_mmru |
| 37 | \( 1 - 2 T + 141 T^{2} - 321 T^{3} + 10015 T^{4} - 21845 T^{5} + 456110 T^{6} - 21845 p T^{7} + 10015 p^{2} T^{8} - 321 p^{3} T^{9} + 141 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.ac_fl_amj_ovf_abgif_zyss |
| 41 | \( 1 - 2 T + 110 T^{2} - 441 T^{3} + 5716 T^{4} - 33392 T^{5} + 236940 T^{6} - 33392 p T^{7} + 5716 p^{2} T^{8} - 441 p^{3} T^{9} + 110 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.ac_eg_aqz_ilw_abxki_nmnc |
| 43 | \( 1 - 10 T + 113 T^{2} - 987 T^{3} + 10213 T^{4} - 67173 T^{5} + 492274 T^{6} - 67173 p T^{7} + 10213 p^{2} T^{8} - 987 p^{3} T^{9} + 113 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ak_ej_ablz_pcv_advjp_bcafq |
| 47 | \( 1 + 14 T + 217 T^{2} + 1312 T^{3} + 9371 T^{4} + 4738 T^{5} + 112278 T^{6} + 4738 p T^{7} + 9371 p^{2} T^{8} + 1312 p^{3} T^{9} + 217 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.o_ij_bym_nwl_hag_gkck |
| 53 | \( 1 - 4 T + 253 T^{2} - 757 T^{3} + 28243 T^{4} - 65007 T^{5} + 1870966 T^{6} - 65007 p T^{7} + 28243 p^{2} T^{8} - 757 p^{3} T^{9} + 253 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.ae_jt_abdd_bpuh_adseh_eclsg |
| 59 | \( 1 + 39 T + 957 T^{2} + 16162 T^{3} + 212911 T^{4} + 2214695 T^{5} + 18887686 T^{6} + 2214695 p T^{7} + 212911 p^{2} T^{8} + 16162 p^{3} T^{9} + 957 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.bn_bkv_xxq_mcyx_ewaep_bpiqjm |
| 61 | \( 1 + 22 T + 387 T^{2} + 4867 T^{3} + 54431 T^{4} + 499255 T^{5} + 4242146 T^{6} + 499255 p T^{7} + 54431 p^{2} T^{8} + 4867 p^{3} T^{9} + 387 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.w_ox_hff_dcnn_bckod_jhjjm |
| 67 | \( 1 - 8 T + 210 T^{2} - 1080 T^{3} + 20055 T^{4} - 70384 T^{5} + 20820 p T^{6} - 70384 p T^{7} + 20055 p^{2} T^{8} - 1080 p^{3} T^{9} + 210 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.ai_ic_abpo_bdrj_aeadc_dbjno |
| 71 | \( 1 + 18 T + 360 T^{2} + 3863 T^{3} + 49660 T^{4} + 423030 T^{5} + 4330398 T^{6} + 423030 p T^{7} + 49660 p^{2} T^{8} + 3863 p^{3} T^{9} + 360 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.s_nw_fsp_cvma_ybuk_jmjxu |
| 73 | \( 1 + 9 T + 379 T^{2} + 3009 T^{3} + 63759 T^{4} + 5761 p T^{5} + 6044588 T^{6} + 5761 p^{2} T^{7} + 63759 p^{2} T^{8} + 3009 p^{3} T^{9} + 379 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.j_op_elt_dqih_xydd_nfxse |
| 79 | \( 1 + 15 T + 226 T^{2} + 1526 T^{3} + 15029 T^{4} + 57511 T^{5} + 861072 T^{6} + 57511 p T^{7} + 15029 p^{2} T^{8} + 1526 p^{3} T^{9} + 226 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.p_is_cgs_wgb_dhbz_bwzue |
| 83 | \( 1 + 43 T + 1151 T^{2} + 21694 T^{3} + 322901 T^{4} + 3880323 T^{5} + 38827446 T^{6} + 3880323 p T^{7} + 322901 p^{2} T^{8} + 21694 p^{3} T^{9} + 1151 p^{4} T^{10} + 43 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.br_bsh_bgck_sjrh_imudf_dgzdbi |
| 89 | \( 1 + 17 T + 418 T^{2} + 3340 T^{3} + 44355 T^{4} + 1271 p T^{5} + 2568076 T^{6} + 1271 p^{2} T^{7} + 44355 p^{2} T^{8} + 3340 p^{3} T^{9} + 418 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.r_qc_eym_cnpz_glit_fqcye |
| 97 | \( 1 - 19 T + 368 T^{2} - 4136 T^{3} + 52959 T^{4} - 457929 T^{5} + 5140904 T^{6} - 457929 p T^{7} + 52959 p^{2} T^{8} - 4136 p^{3} T^{9} + 368 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.at_oe_agdc_daix_ababkr_lgmxc |
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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
−5.14900372679214825747502399748, −4.74651889980784436446304704280, −4.70639212953795368502367923327, −4.65206871622392433311763812353, −4.60000217342846184519729989834, −4.48194620778017743141920999165, −4.41415079810112398528826875884, −4.12739405520044860020741282036, −3.83826149579392056951730324511, −3.79181305607087287151801061700, −3.61412843080407101279044206144, −3.58046953537691642365797446483, −3.13716253777292750597902857489, −3.11764791117904080875007769210, −2.89389397621718255751563942071, −2.79632091215673012273795461562, −2.71510154327812049327345348550, −2.50537057447287923327387678490, −2.12912243375545003101016481759, −2.01776509400075094032012958850, −1.72193358405821942957389150410, −1.58361177908080091887515365427, −1.31109386111989722966913447678, −1.25765097437469950246998529509, −1.06852986206175635826927411006, 0, 0, 0, 0, 0, 0,
1.06852986206175635826927411006, 1.25765097437469950246998529509, 1.31109386111989722966913447678, 1.58361177908080091887515365427, 1.72193358405821942957389150410, 2.01776509400075094032012958850, 2.12912243375545003101016481759, 2.50537057447287923327387678490, 2.71510154327812049327345348550, 2.79632091215673012273795461562, 2.89389397621718255751563942071, 3.11764791117904080875007769210, 3.13716253777292750597902857489, 3.58046953537691642365797446483, 3.61412843080407101279044206144, 3.79181305607087287151801061700, 3.83826149579392056951730324511, 4.12739405520044860020741282036, 4.41415079810112398528826875884, 4.48194620778017743141920999165, 4.60000217342846184519729989834, 4.65206871622392433311763812353, 4.70639212953795368502367923327, 4.74651889980784436446304704280, 5.14900372679214825747502399748
Plot not available for L-functions of degree greater than 10.
