| L(s) = 1 | + 3·3-s + 2·5-s − 3·7-s + 6·9-s + 2·11-s + 2·13-s + 6·15-s + 3·17-s − 2·19-s − 9·21-s − 2·23-s + 10·27-s − 6·31-s + 6·33-s − 6·35-s + 8·37-s + 6·39-s + 2·41-s + 6·43-s + 12·45-s + 6·47-s + 6·49-s + 9·51-s + 12·53-s + 4·55-s − 6·57-s + 6·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 1.13·7-s + 2·9-s + 0.603·11-s + 0.554·13-s + 1.54·15-s + 0.727·17-s − 0.458·19-s − 1.96·21-s − 0.417·23-s + 1.92·27-s − 1.07·31-s + 1.04·33-s − 1.01·35-s + 1.31·37-s + 0.960·39-s + 0.312·41-s + 0.914·43-s + 1.78·45-s + 0.875·47-s + 6/7·49-s + 1.26·51-s + 1.64·53-s + 0.539·55-s − 0.794·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.18400283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.18400283\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 4 T^{2} - 2 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_e_ac |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 2 T^{2} + 24 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_c_y |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 28 T^{2} - 34 T^{3} + 28 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bc_abi |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 46 T^{2} + 58 T^{3} + 46 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_bu_cg |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 38 T^{2} + 24 T^{3} + 38 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_bm_y |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.29.a_dj_a |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 228 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.g_cl_iu |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 388 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_dh_aoy |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 112 T^{2} - 146 T^{3} + 112 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_ei_afq |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 54 T^{2} - 652 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ag_cc_azc |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 111 T^{2} - 420 T^{3} + 111 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_eh_aqe |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1236 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.am_gj_abvo |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 147 T^{2} - 700 T^{3} + 147 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_fr_abay |
| 61 | $S_4\times C_2$ | \( 1 - 22 T + 297 T^{2} - 2828 T^{3} + 297 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.aw_ll_aeeu |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 285 T^{2} + 2728 T^{3} + 285 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.u_kz_eay |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 169 T^{2} - 424 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ae_gn_aqi |
| 73 | $S_4\times C_2$ | \( 1 - 20 T + 303 T^{2} - 2968 T^{3} + 303 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.au_lr_aeke |
| 79 | $S_4\times C_2$ | \( 1 + 22 T + 351 T^{2} + 3620 T^{3} + 351 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.w_nn_fjg |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 51 T^{2} + 1752 T^{3} - 51 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.am_abz_cpk |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 219 T^{2} - 404 T^{3} + 219 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ac_il_apo |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) | 3.97.ay_sp_ahqu |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.23677986837656151300277255472, −7.00373321164167755621269005943, −6.71815036695142253309856592207, −6.60162019092580701625079821740, −6.07224318790314611784529524657, −5.99036570433182612619236253075, −5.94595443951905092702311875981, −5.56385142844466155126804891513, −5.28976665791959174381744058437, −5.05984196743849424408070752838, −4.52137442309255004735084822002, −4.27639429410634273440031586116, −4.17577892027018776573049848737, −3.78569480383237448354113177790, −3.61898778037838371471982014467, −3.47712125390292615719279828047, −3.05899150074664802670327234000, −2.70821098686964112951444639403, −2.63486131424806015452320286451, −2.12460330677081252212610924649, −1.93309420045725704823706897419, −1.87009645526486604795663499504, −1.14284112860343579283185559351, −0.805066516087515224214994694965, −0.60545710900837815185082836550,
0.60545710900837815185082836550, 0.805066516087515224214994694965, 1.14284112860343579283185559351, 1.87009645526486604795663499504, 1.93309420045725704823706897419, 2.12460330677081252212610924649, 2.63486131424806015452320286451, 2.70821098686964112951444639403, 3.05899150074664802670327234000, 3.47712125390292615719279828047, 3.61898778037838371471982014467, 3.78569480383237448354113177790, 4.17577892027018776573049848737, 4.27639429410634273440031586116, 4.52137442309255004735084822002, 5.05984196743849424408070752838, 5.28976665791959174381744058437, 5.56385142844466155126804891513, 5.94595443951905092702311875981, 5.99036570433182612619236253075, 6.07224318790314611784529524657, 6.60162019092580701625079821740, 6.71815036695142253309856592207, 7.00373321164167755621269005943, 7.23677986837656151300277255472
