| L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 3·5-s − 3·6-s + 4·7-s + 10·8-s − 4·9-s − 9·10-s − 5·11-s − 6·12-s − 13-s + 12·14-s + 3·15-s + 15·16-s + 17-s − 12·18-s + 3·19-s − 18·20-s − 4·21-s − 15·22-s − 5·23-s − 10·24-s + 6·25-s − 3·26-s + 4·27-s + 24·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 3·4-s − 1.34·5-s − 1.22·6-s + 1.51·7-s + 3.53·8-s − 4/3·9-s − 2.84·10-s − 1.50·11-s − 1.73·12-s − 0.277·13-s + 3.20·14-s + 0.774·15-s + 15/4·16-s + 0.242·17-s − 2.82·18-s + 0.688·19-s − 4.02·20-s − 0.872·21-s − 3.19·22-s − 1.04·23-s − 2.04·24-s + 6/5·25-s − 0.588·26-s + 0.769·27-s + 4.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 31 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_f_f |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 22 T^{2} - 53 T^{3} + 22 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_w_acb |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 15 T^{2} + 29 T^{3} + 15 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.f_p_bd |
| 13 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 59 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.13.b_n_ch |
| 17 | $S_4\times C_2$ | \( 1 - T + 45 T^{2} - 37 T^{3} + 45 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ab_bt_abl |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_bz_aed |
| 23 | $S_4\times C_2$ | \( 1 + 5 T + 3 T^{2} - 133 T^{3} + 3 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.f_d_afd |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 24 T^{2} - 31 T^{3} + 24 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_y_abf |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 65 T^{2} + 173 T^{3} + 65 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.f_cn_gr |
| 41 | $S_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 327 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ad_cr_amp |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 45 T^{2} - 115 T^{3} + 45 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_bt_ael |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 144 T^{2} + 555 T^{3} + 144 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_fo_vj |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 93 T^{2} + 587 T^{3} + 93 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.l_dp_wp |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 156 T^{2} + 1147 T^{3} + 156 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.k_ga_bsd |
| 61 | $S_4\times C_2$ | \( 1 + 7 T + 130 T^{2} + 731 T^{3} + 130 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.h_fa_bcd |
| 67 | $S_4\times C_2$ | \( 1 + 25 T + 392 T^{2} + 3781 T^{3} + 392 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.z_pc_fpl |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_hz_abub |
| 73 | $S_4\times C_2$ | \( 1 + 28 T + 430 T^{2} + 4319 T^{3} + 430 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.bc_qo_gkd |
| 79 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1141 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aj_hz_abrx |
| 83 | $S_4\times C_2$ | \( 1 - 3 T + 45 T^{2} - 975 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ad_bt_abln |
| 89 | $S_4\times C_2$ | \( 1 + 9 T + 213 T^{2} + 1359 T^{3} + 213 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.j_if_cah |
| 97 | $S_4\times C_2$ | \( 1 + 16 T + 356 T^{2} + 3181 T^{3} + 356 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.q_ns_esj |
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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
−7.32302549309527222860475459706, −6.78304271035749549650175955342, −6.51008950918815018007231855221, −6.33492396796234264571805879511, −5.96619446181234551538993534426, −5.90431599099670713202556126143, −5.77295753750154418158172166291, −5.33798032863462403237401806011, −5.22148010060998251969293979484, −5.12237639423742867816432161961, −4.71666017279684085294307734115, −4.60499486435908382866792727875, −4.50334785519648693354002626260, −4.14615316049259245503868115653, −4.01054606301844634650195458869, −3.45026171291091363585309208159, −3.25745693472834006663790484867, −3.16862782362472282922175495472, −3.02575977571629989826404001750, −2.49442651717270500542698239840, −2.39784783423767701680006197057, −2.11104875697948212403857928999, −1.53930576856671913362345692817, −1.33530144208802741361949739036, −1.18208208314719036169747992880, 0, 0, 0,
1.18208208314719036169747992880, 1.33530144208802741361949739036, 1.53930576856671913362345692817, 2.11104875697948212403857928999, 2.39784783423767701680006197057, 2.49442651717270500542698239840, 3.02575977571629989826404001750, 3.16862782362472282922175495472, 3.25745693472834006663790484867, 3.45026171291091363585309208159, 4.01054606301844634650195458869, 4.14615316049259245503868115653, 4.50334785519648693354002626260, 4.60499486435908382866792727875, 4.71666017279684085294307734115, 5.12237639423742867816432161961, 5.22148010060998251969293979484, 5.33798032863462403237401806011, 5.77295753750154418158172166291, 5.90431599099670713202556126143, 5.96619446181234551538993534426, 6.33492396796234264571805879511, 6.51008950918815018007231855221, 6.78304271035749549650175955342, 7.32302549309527222860475459706
