| L(s) = 1 | − 3·3-s − 3·7-s + 6·9-s − 2·11-s + 10·13-s + 8·17-s − 2·19-s + 9·21-s + 8·23-s − 10·27-s − 6·29-s − 14·31-s + 6·33-s + 12·37-s − 30·39-s − 10·41-s + 4·43-s + 6·49-s − 24·51-s + 14·53-s + 6·57-s − 8·59-s + 2·61-s − 18·63-s − 16·67-s − 24·69-s − 18·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 2·9-s − 0.603·11-s + 2.77·13-s + 1.94·17-s − 0.458·19-s + 1.96·21-s + 1.66·23-s − 1.92·27-s − 1.11·29-s − 2.51·31-s + 1.04·33-s + 1.97·37-s − 4.80·39-s − 1.56·41-s + 0.609·43-s + 6/7·49-s − 3.36·51-s + 1.92·53-s + 0.794·57-s − 1.04·59-s + 0.256·61-s − 2.26·63-s − 1.95·67-s − 2.88·69-s − 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.433306668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433306668\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 11 | $S_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 60 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_ad_aci |
| 13 | $S_4\times C_2$ | \( 1 - 10 T + 59 T^{2} - 252 T^{3} + 59 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ak_ch_ajs |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 19 T^{2} + 19 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ai_t_a |
| 19 | $S_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 124 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_ad_aeu |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 29 T^{2} - 64 T^{3} + 29 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ai_bd_acm |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.29.g_dv_ns |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 145 T^{2} + 908 T^{3} + 145 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.o_fp_biy |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 95 T^{2} - 568 T^{3} + 95 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.am_dr_avw |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_fn_bfg |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 81 T^{2} - 280 T^{3} + 81 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ae_dd_aku |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.47.a_fl_a |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 211 T^{2} - 1524 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ao_id_acgq |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.i_ej_bam |
| 61 | $D_{6}$ | \( 1 - 2 T - 29 T^{2} - 140 T^{3} - 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ac_abd_afk |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 3 p T^{2} + 1888 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.q_ht_cuq |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 161 T^{2} + 1204 T^{3} + 161 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.s_gf_bui |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 319 T^{2} - 3316 T^{3} + 319 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aw_mh_aexo |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 189 T^{2} - 568 T^{3} + 189 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_hh_avw |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ai_hd_abpg |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 143 T^{2} + 1300 T^{3} + 143 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_fn_bya |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 231 T^{2} - 188 T^{3} + 231 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ac_ix_ahg |
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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
−7.39742336360968083806906143127, −7.04695937630985909181631309473, −7.03694358958118023693915619967, −6.64346641880773483296526408942, −6.28729919496132713418942660297, −6.14392777364396821979787678204, −5.85318139611784898458165990504, −5.67076961193174245177174276128, −5.63281584803271297240038750259, −5.50721772760819694669428160267, −4.89613501185534754222989723778, −4.65782248287610845026908456887, −4.58028315804638050865538773773, −4.00865267580996968367031008428, −3.74395382566675955436993015259, −3.60537822369412591665978499565, −3.26199858446333215858434490670, −3.17575882546213817440205683765, −2.80203674624821573858188416216, −2.04914929174397113148928453626, −1.83231529719892780412328532312, −1.56219300253496594831719765406, −0.963907712723982093926808996824, −0.67493250567290011178770288893, −0.52470202096545724293687002258,
0.52470202096545724293687002258, 0.67493250567290011178770288893, 0.963907712723982093926808996824, 1.56219300253496594831719765406, 1.83231529719892780412328532312, 2.04914929174397113148928453626, 2.80203674624821573858188416216, 3.17575882546213817440205683765, 3.26199858446333215858434490670, 3.60537822369412591665978499565, 3.74395382566675955436993015259, 4.00865267580996968367031008428, 4.58028315804638050865538773773, 4.65782248287610845026908456887, 4.89613501185534754222989723778, 5.50721772760819694669428160267, 5.63281584803271297240038750259, 5.67076961193174245177174276128, 5.85318139611784898458165990504, 6.14392777364396821979787678204, 6.28729919496132713418942660297, 6.64346641880773483296526408942, 7.03694358958118023693915619967, 7.04695937630985909181631309473, 7.39742336360968083806906143127
