| L(s) = 1 | + 2·3-s + 3·5-s + 4·7-s − 3·11-s − 6·13-s + 6·15-s + 2·17-s + 6·19-s + 8·21-s + 6·23-s + 6·25-s + 2·27-s + 6·29-s − 6·33-s + 12·35-s − 16·37-s − 12·39-s + 6·41-s + 2·43-s + 6·47-s + 2·49-s + 4·51-s − 16·53-s − 9·55-s + 12·57-s − 4·59-s − 14·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s + 1.51·7-s − 0.904·11-s − 1.66·13-s + 1.54·15-s + 0.485·17-s + 1.37·19-s + 1.74·21-s + 1.25·23-s + 6/5·25-s + 0.384·27-s + 1.11·29-s − 1.04·33-s + 2.02·35-s − 2.63·37-s − 1.92·39-s + 0.937·41-s + 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.560·51-s − 2.19·53-s − 1.21·55-s + 1.58·57-s − 0.520·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.59050985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.59050985\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 4 T^{2} - 10 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_e_ak |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 2 p T^{2} - 30 T^{3} + 2 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_o_abe |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.13.g_bz_gi |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 46 T^{2} - 66 T^{3} + 46 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ac_bu_aco |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 26 T^{2} - 44 T^{3} + 26 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ag_ba_abs |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.23.ag_dd_aky |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} - 266 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_bm_akg |
| 31 | $S_4\times C_2$ | \( 1 + 36 T^{2} + 52 T^{3} + 36 p T^{4} + p^{3} T^{6} \) | 3.31.a_bk_ca |
| 37 | $S_4\times C_2$ | \( 1 + 16 T + 184 T^{2} + 34 p T^{3} + 184 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.q_hc_bwk |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ct_aqu |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + T^{2} + 340 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_b_nc |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 508 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_dl_ato |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 232 T^{2} + 1770 T^{3} + 232 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.q_iy_cqc |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 728 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.e_cb_bca |
| 61 | $S_4\times C_2$ | \( 1 + 14 T + 198 T^{2} + 1706 T^{3} + 198 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.o_hq_cnq |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 217 T^{2} - 1652 T^{3} + 217 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ao_ij_aclo |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 228 T^{2} + 1140 T^{3} + 228 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.i_iu_brw |
| 73 | $S_4\times C_2$ | \( 1 - 26 T + 359 T^{2} - 3468 T^{3} + 359 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aba_nv_afdk |
| 79 | $S_4\times C_2$ | \( 1 - 32 T + 7 p T^{2} - 5984 T^{3} + 7 p^{2} T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.abg_vh_aiwe |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 197 T^{2} - 1652 T^{3} + 197 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ak_hp_aclo |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 152 T^{2} - 146 T^{3} + 152 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ae_fw_afq |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 91 T^{2} - 796 T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.c_dn_abeq |
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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
−7.65540771790903723568913479278, −7.48130304391841623912273950826, −7.27021047483807718880107826256, −6.83719279526218875219857383784, −6.64342570725449577774100684539, −6.37520423429113158664461885175, −6.12891703578275676205515922230, −5.72365883574909034121643647419, −5.38686203603944106818627165707, −5.17100283447274733885428980500, −4.89336636661732147230659139216, −4.88409026600304492902713694538, −4.85721093788513368174288819929, −4.36030376987323962661761005586, −3.70499300405858881548550864907, −3.48044015008761845592412236635, −3.14718655998096588491028575283, −2.93911214213540648534402578205, −2.77822684605414521538493732884, −2.27685357566315517708518757053, −1.99330159827200088562508988077, −1.94347928915631018885644020369, −1.44310054328567771814257239482, −0.76888392142423475582415188073, −0.70266572627259826332942330424,
0.70266572627259826332942330424, 0.76888392142423475582415188073, 1.44310054328567771814257239482, 1.94347928915631018885644020369, 1.99330159827200088562508988077, 2.27685357566315517708518757053, 2.77822684605414521538493732884, 2.93911214213540648534402578205, 3.14718655998096588491028575283, 3.48044015008761845592412236635, 3.70499300405858881548550864907, 4.36030376987323962661761005586, 4.85721093788513368174288819929, 4.88409026600304492902713694538, 4.89336636661732147230659139216, 5.17100283447274733885428980500, 5.38686203603944106818627165707, 5.72365883574909034121643647419, 6.12891703578275676205515922230, 6.37520423429113158664461885175, 6.64342570725449577774100684539, 6.83719279526218875219857383784, 7.27021047483807718880107826256, 7.48130304391841623912273950826, 7.65540771790903723568913479278
