| L(s) = 1 | + 3-s − 5-s + 4·7-s − 5·9-s − 5·11-s − 4·13-s − 15-s + 2·17-s − 3·19-s + 4·21-s + 14·23-s − 8·25-s − 4·27-s − 3·29-s + 6·31-s − 5·33-s − 4·35-s − 37-s − 4·39-s + 9·41-s + 5·43-s + 5·45-s + 14·47-s + 10·49-s + 2·51-s + 53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s − 5/3·9-s − 1.50·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s − 0.688·19-s + 0.872·21-s + 2.91·23-s − 8/5·25-s − 0.769·27-s − 0.557·29-s + 1.07·31-s − 0.870·33-s − 0.676·35-s − 0.164·37-s − 0.640·39-s + 1.40·41-s + 0.762·43-s + 0.745·45-s + 2.04·47-s + 10/7·49-s + 0.280·51-s + 0.137·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.353032996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.353032996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} - 7 T^{3} + 20 T^{4} - 7 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ab_g_ah_u |
| 5 | $C_2 \wr S_4$ | \( 1 + T + 9 T^{2} + 6 T^{3} + 12 p T^{4} + 6 p T^{5} + 9 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.5.b_j_g_ci |
| 11 | $C_2 \wr S_4$ | \( 1 + 5 T + 42 T^{2} + 141 T^{3} + 684 T^{4} + 141 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.f_bq_fl_bai |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 40 T^{2} - 8 p T^{3} + 774 T^{4} - 8 p^{2} T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ac_bo_afg_bdu |
| 19 | $C_2 \wr S_4$ | \( 1 + 3 T + 51 T^{2} + 124 T^{3} + 1374 T^{4} + 124 p T^{5} + 51 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.d_bz_eu_caw |
| 23 | $C_2 \wr S_4$ | \( 1 - 14 T + 140 T^{2} - 968 T^{3} + 5285 T^{4} - 968 p T^{5} + 140 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ao_fk_ablg_hvh |
| 29 | $C_2 \wr S_4$ | \( 1 + 3 T + 67 T^{2} + 30 T^{3} + 1992 T^{4} + 30 p T^{5} + 67 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.d_cp_be_cyq |
| 31 | $C_2 \wr S_4$ | \( 1 - 6 T + 108 T^{2} - 546 T^{3} + 4775 T^{4} - 546 p T^{5} + 108 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ag_ee_ava_hbr |
| 37 | $C_2 \wr S_4$ | \( 1 + T + 50 T^{2} + 39 T^{3} + 952 T^{4} + 39 p T^{5} + 50 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.37.b_by_bn_bkq |
| 41 | $C_2 \wr S_4$ | \( 1 - 9 T + 142 T^{2} - 1071 T^{3} + 8322 T^{4} - 1071 p T^{5} + 142 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.aj_fm_abpf_mic |
| 43 | $C_2 \wr S_4$ | \( 1 - 5 T + 119 T^{2} - 732 T^{3} + 6500 T^{4} - 732 p T^{5} + 119 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.af_ep_abce_jqa |
| 47 | $C_2 \wr S_4$ | \( 1 - 14 T + 90 T^{2} - 174 T^{3} - 87 T^{4} - 174 p T^{5} + 90 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ao_dm_ags_adj |
| 53 | $C_2 \wr S_4$ | \( 1 - T + 187 T^{2} - 98 T^{3} + 14172 T^{4} - 98 p T^{5} + 187 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ab_hf_adu_uzc |
| 59 | $C_2 \wr S_4$ | \( 1 - 4 T + 144 T^{2} - 192 T^{3} + 9670 T^{4} - 192 p T^{5} + 144 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ae_fo_ahk_ohy |
| 61 | $C_2 \wr S_4$ | \( 1 - 11 T + 242 T^{2} - 1857 T^{3} + 22010 T^{4} - 1857 p T^{5} + 242 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.al_ji_actl_bgoo |
| 67 | $C_2 \wr S_4$ | \( 1 - 9 T + 274 T^{2} - 1797 T^{3} + 27730 T^{4} - 1797 p T^{5} + 274 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.aj_ko_acrd_bpao |
| 71 | $C_2 \wr S_4$ | \( 1 - 16 T + 170 T^{2} - 1006 T^{3} + 118 p T^{4} - 1006 p T^{5} + 170 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aq_go_abms_mkg |
| 73 | $C_2 \wr S_4$ | \( 1 - 2 T + 184 T^{2} - 842 T^{3} + 15827 T^{4} - 842 p T^{5} + 184 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ac_hc_abgk_xkt |
| 79 | $C_2 \wr S_4$ | \( 1 - 20 T + 370 T^{2} - 4124 T^{3} + 43847 T^{4} - 4124 p T^{5} + 370 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.au_og_agcq_cmwl |
| 83 | $C_2 \wr S_4$ | \( 1 + T + 231 T^{2} - 318 T^{3} + 23740 T^{4} - 318 p T^{5} + 231 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.83.b_ix_amg_bjdc |
| 89 | $C_2 \wr S_4$ | \( 1 + 7 T + 317 T^{2} + 1720 T^{3} + 40642 T^{4} + 1720 p T^{5} + 317 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.h_mf_coe_cide |
| 97 | $C_2 \wr S_4$ | \( 1 + 282 T^{2} + 342 T^{3} + 35837 T^{4} + 342 p T^{5} + 282 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_kw_ne_cbaj |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.35955134884239822907790927406, −5.69475295511928048959679409286, −5.60558924753562876895091383182, −5.52270792796110209625456689401, −5.49895407975009660368436596733, −5.20950320741758270083316141815, −5.02619267347960069075830186402, −4.90950429978456359534572123279, −4.62465118060521321797523166063, −4.22045969887527592109052284957, −4.08544452797038866617627661397, −3.96858376887168227822206104863, −3.87237696206298146771367854463, −3.22523072681524281137763706300, −3.14118770415493542731881493452, −2.89007698645036976801423071678, −2.81236760188766593426587547068, −2.37396617087539728814374899433, −2.32599706237977136509438582544, −2.10455132350222132473585463286, −1.95330914988748865520005035741, −1.24718275957764489410533684468, −0.964475783983458357940089570103, −0.64457666917570679093801003541, −0.39640626635229236424298523488,
0.39640626635229236424298523488, 0.64457666917570679093801003541, 0.964475783983458357940089570103, 1.24718275957764489410533684468, 1.95330914988748865520005035741, 2.10455132350222132473585463286, 2.32599706237977136509438582544, 2.37396617087539728814374899433, 2.81236760188766593426587547068, 2.89007698645036976801423071678, 3.14118770415493542731881493452, 3.22523072681524281137763706300, 3.87237696206298146771367854463, 3.96858376887168227822206104863, 4.08544452797038866617627661397, 4.22045969887527592109052284957, 4.62465118060521321797523166063, 4.90950429978456359534572123279, 5.02619267347960069075830186402, 5.20950320741758270083316141815, 5.49895407975009660368436596733, 5.52270792796110209625456689401, 5.60558924753562876895091383182, 5.69475295511928048959679409286, 6.35955134884239822907790927406
