| L(s) = 1 | − 2·4-s − 4·7-s − 8-s + 2·11-s − 4·13-s − 2·17-s + 4·19-s − 6·23-s + 8·28-s + 8·29-s − 3·31-s + 4·32-s + 10·41-s − 14·43-s − 4·44-s + 12·47-s − 4·49-s + 8·52-s − 10·53-s + 4·56-s − 26·59-s − 2·61-s + 64-s − 12·67-s + 4·68-s + 10·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s − 0.353·8-s + 0.603·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s − 1.25·23-s + 1.51·28-s + 1.48·29-s − 0.538·31-s + 0.707·32-s + 1.56·41-s − 2.13·43-s − 0.603·44-s + 1.75·47-s − 4/7·49-s + 1.10·52-s − 1.37·53-s + 0.534·56-s − 3.38·59-s − 0.256·61-s + 1/8·64-s − 1.46·67-s + 0.485·68-s + 1.18·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8268724443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8268724443\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{6} \) | 3.2.a_c_b |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 48 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_u_bw |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 60 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_n_aci |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 48 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.e_x_bw |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 36 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_bb_bk |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 12 T^{2} + 44 T^{3} + 12 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ae_m_bs |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 65 T^{2} + 244 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cn_jk |
| 29 | $S_4\times C_2$ | \( 1 - 8 T + 31 T^{2} - 72 T^{3} + 31 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ai_bf_acu |
| 37 | $S_4\times C_2$ | \( 1 + 95 T^{2} - 8 T^{3} + 95 p T^{4} + p^{3} T^{6} \) | 3.37.a_dr_ai |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 106 T^{2} - 558 T^{3} + 106 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ak_ec_avm |
| 43 | $S_4\times C_2$ | \( 1 + 14 T + 133 T^{2} + 836 T^{3} + 133 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.o_fd_bge |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 125 T^{2} - 872 T^{3} + 125 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.am_ev_abho |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 143 T^{2} + 1028 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.k_fn_bno |
| 59 | $S_4\times C_2$ | \( 1 + 26 T + 390 T^{2} + 3624 T^{3} + 390 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ba_pa_fjk |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 55 T^{2} - 268 T^{3} + 55 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_cd_aki |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) | 3.67.m_jp_cmi |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 66 T^{2} - 708 T^{3} + 66 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ak_co_abbg |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 123 T^{2} - 1024 T^{3} + 123 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_et_abnk |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ai_iz_abue |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 357 T^{2} - 3432 T^{3} + 357 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.au_nt_afca |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) | 3.89.as_ol_afbo |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 264 T^{2} + 682 T^{3} + 264 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_ke_bag |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.09653967126336739674342834490, −6.58310639679835580118633930623, −6.38219466273425013337663854294, −6.35610277182059350494754082151, −6.22553176725656855273204646936, −6.04437049493956624344549633912, −5.68188824916690826804097800028, −5.11264749747996218789899952738, −5.02990455691655653753462441860, −4.94229544388264232512547455536, −4.61499896770718643163117201915, −4.51281282045657624576361711859, −3.95327724874915863987378421501, −3.87689563857915707150153234316, −3.64156854913907637614807671686, −3.19932311150282099717446557411, −3.05211750546090790382073863253, −2.93693826374687501059818649068, −2.47402958799522862068702799391, −2.17305335546581297213914537948, −1.68425277648219596072722662983, −1.66245729128406892866788730603, −0.841156521592552853533878514791, −0.58906712269760085240916219558, −0.24177327137074498608443555317,
0.24177327137074498608443555317, 0.58906712269760085240916219558, 0.841156521592552853533878514791, 1.66245729128406892866788730603, 1.68425277648219596072722662983, 2.17305335546581297213914537948, 2.47402958799522862068702799391, 2.93693826374687501059818649068, 3.05211750546090790382073863253, 3.19932311150282099717446557411, 3.64156854913907637614807671686, 3.87689563857915707150153234316, 3.95327724874915863987378421501, 4.51281282045657624576361711859, 4.61499896770718643163117201915, 4.94229544388264232512547455536, 5.02990455691655653753462441860, 5.11264749747996218789899952738, 5.68188824916690826804097800028, 6.04437049493956624344549633912, 6.22553176725656855273204646936, 6.35610277182059350494754082151, 6.38219466273425013337663854294, 6.58310639679835580118633930623, 7.09653967126336739674342834490
