| L(s) = 1 | − 3-s + 3·5-s − 3·7-s − 3·9-s − 6·11-s + 13-s − 3·15-s + 5·17-s − 2·19-s + 3·21-s + 11·23-s + 6·25-s − 27-s + 3·29-s + 31-s + 6·33-s − 9·35-s − 14·37-s − 39-s − 19·43-s − 9·45-s + 10·47-s − 11·49-s − 5·51-s − 9·53-s − 18·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1.13·7-s − 9-s − 1.80·11-s + 0.277·13-s − 0.774·15-s + 1.21·17-s − 0.458·19-s + 0.654·21-s + 2.29·23-s + 6/5·25-s − 0.192·27-s + 0.557·29-s + 0.179·31-s + 1.04·33-s − 1.52·35-s − 2.30·37-s − 0.160·39-s − 2.89·43-s − 1.34·45-s + 1.45·47-s − 1.57·49-s − 0.700·51-s − 1.23·53-s − 2.42·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + 8 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_e_i |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 20 T^{2} + 38 T^{3} + 20 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_u_bm |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 29 T^{2} + 100 T^{3} + 29 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bd_dw |
| 13 | $S_4\times C_2$ | \( 1 - T + 34 T^{2} - 28 T^{3} + 34 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ab_bi_abc |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 28 T^{2} - 114 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.af_bc_aek |
| 19 | $S_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 148 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_ad_afs |
| 23 | $S_4\times C_2$ | \( 1 - 11 T + 94 T^{2} - 522 T^{3} + 94 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.al_dq_auc |
| 31 | $S_4\times C_2$ | \( 1 - T + 74 T^{2} - 30 T^{3} + 74 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ab_cw_abe |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 155 T^{2} + 1068 T^{3} + 155 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.o_fz_bpc |
| 41 | $S_4\times C_2$ | \( 1 + 107 T^{2} + 8 T^{3} + 107 p T^{4} + p^{3} T^{6} \) | 3.41.a_ed_i |
| 43 | $S_4\times C_2$ | \( 1 + 19 T + 230 T^{2} + 1740 T^{3} + 230 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.t_iw_coy |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 149 T^{2} - 908 T^{3} + 149 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ak_ft_abiy |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 150 T^{2} + 900 T^{3} + 150 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_fu_biq |
| 59 | $S_4\times C_2$ | \( 1 + T + 40 T^{2} - 6 T^{3} + 40 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.59.b_bo_ag |
| 61 | $S_4\times C_2$ | \( 1 + 7 T + 68 T^{2} + 420 T^{3} + 68 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.h_cq_qe |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 121 T^{2} + 944 T^{3} + 121 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_er_bki |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 197 T^{2} - 1760 T^{3} + 197 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aq_hp_acps |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 68 T^{2} + 82 T^{3} + 68 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_cq_de |
| 79 | $S_4\times C_2$ | \( 1 - 7 T + 196 T^{2} - 898 T^{3} + 196 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ah_ho_abio |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 261 T^{2} + 1596 T^{3} + 261 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.k_kb_cjk |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 379 T^{2} + 3744 T^{3} + 379 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_op_foa |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 236 T^{2} + 1634 T^{3} + 236 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.j_jc_ckw |
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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
−7.06931705533546717400160870790, −6.75846383300952755116416240294, −6.56311255125387578210757106787, −6.55128283777615847243449456659, −6.27094147079316393498732949817, −5.91209017111861440109587376267, −5.83051197014737555545888944799, −5.40638335675322878297960812709, −5.38528533146455071584589969437, −5.15538697162582177026159231868, −4.94924088415403238007292198714, −4.85475572117772338141792895841, −4.58870731764291538585096154062, −3.90830726588725094383852845775, −3.76713507127828806793573984563, −3.45354308101898484161962314593, −3.17684655045528539998396203416, −2.94431060199252045590599072200, −2.92001740244412000876615524916, −2.60841566671879364054800256876, −2.18242846244418073905981824819, −1.91974995522407848572741257978, −1.56175105858447433815669359037, −1.14160779480505134322744603382, −1.01682144007581638035205575672, 0, 0, 0,
1.01682144007581638035205575672, 1.14160779480505134322744603382, 1.56175105858447433815669359037, 1.91974995522407848572741257978, 2.18242846244418073905981824819, 2.60841566671879364054800256876, 2.92001740244412000876615524916, 2.94431060199252045590599072200, 3.17684655045528539998396203416, 3.45354308101898484161962314593, 3.76713507127828806793573984563, 3.90830726588725094383852845775, 4.58870731764291538585096154062, 4.85475572117772338141792895841, 4.94924088415403238007292198714, 5.15538697162582177026159231868, 5.38528533146455071584589969437, 5.40638335675322878297960812709, 5.83051197014737555545888944799, 5.91209017111861440109587376267, 6.27094147079316393498732949817, 6.55128283777615847243449456659, 6.56311255125387578210757106787, 6.75846383300952755116416240294, 7.06931705533546717400160870790
