| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521762496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521762496\) |
| \(L(\frac{3}{2})\) |
|
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.c_e_k |
| 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.ag_bz_agi |
| 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.g_ba_bs |
| 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.g_bm_kg |
| 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.aq_hc_abwk |
| 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.c_b_anc |
| 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.aq_iy_acqc |
| 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.ae_cb_abca |
| 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.ao_hq_acnq |
| 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.o_ij_clo |
| 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.k_hp_clo |
| 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 |
| 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
−8.394792940394789782348324038834, −7.81939911910218389294850292202, −7.75937597442256877732494017545, −7.52120532915856866596012422251, −7.36740843772507725200038489203, −6.79806456793902197155631296331, −6.61411953468297327804793379968, −6.31664214281348504385483919723, −6.01837488905012859786558357754, −5.96143448646420481152953462231, −5.32610906721128776071438260764, −5.30191423182351307613870771780, −5.09272763662581174281828191471, −4.48196943626907437611568600541, −4.21684673938915490866941438235, −4.21434997029737236012357487863, −3.81385771608457819062765038405, −3.37085049062580452456013508926, −3.36063950981497605195535292646, −2.45528127663243067120428585020, −2.30011973196866181789901205095, −1.82012445723443129568662477477, −1.21386094546358552502413362974, −0.74181365708125352991483581524, −0.68245803210376181766453423790,
0.68245803210376181766453423790, 0.74181365708125352991483581524, 1.21386094546358552502413362974, 1.82012445723443129568662477477, 2.30011973196866181789901205095, 2.45528127663243067120428585020, 3.36063950981497605195535292646, 3.37085049062580452456013508926, 3.81385771608457819062765038405, 4.21434997029737236012357487863, 4.21684673938915490866941438235, 4.48196943626907437611568600541, 5.09272763662581174281828191471, 5.30191423182351307613870771780, 5.32610906721128776071438260764, 5.96143448646420481152953462231, 6.01837488905012859786558357754, 6.31664214281348504385483919723, 6.61411953468297327804793379968, 6.79806456793902197155631296331, 7.36740843772507725200038489203, 7.52120532915856866596012422251, 7.75937597442256877732494017545, 7.81939911910218389294850292202, 8.394792940394789782348324038834
