| L(s) = 1 | − 3·3-s + 5·5-s − 3·7-s + 9-s + 2·13-s − 15·15-s − 3·17-s − 2·19-s + 9·21-s + 4·23-s + 7·25-s + 5·27-s + 10·29-s − 7·31-s − 15·35-s + 22·37-s − 6·39-s − 11·41-s + 7·43-s + 5·45-s + 12·47-s + 6·49-s + 9·51-s + 9·53-s + 6·57-s − 8·59-s + 27·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2.23·5-s − 1.13·7-s + 1/3·9-s + 0.554·13-s − 3.87·15-s − 0.727·17-s − 0.458·19-s + 1.96·21-s + 0.834·23-s + 7/5·25-s + 0.962·27-s + 1.85·29-s − 1.25·31-s − 2.53·35-s + 3.61·37-s − 0.960·39-s − 1.71·41-s + 1.06·43-s + 0.745·45-s + 1.75·47-s + 6/7·49-s + 1.26·51-s + 1.23·53-s + 0.794·57-s − 1.04·59-s + 3.45·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054809609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054809609\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 8 T^{2} + 16 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_i_q |
| 5 | $S_4\times C_2$ | \( 1 - p T + 18 T^{2} - 42 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.5.af_s_abq |
| 11 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 8 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.11.a_r_ai |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 20 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_p_au |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 92 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_bl_do |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 168 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_bx_agm |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 516 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ak_dv_atw |
| 31 | $S_4\times C_2$ | \( 1 + 7 T + 52 T^{2} + 226 T^{3} + 52 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.h_ca_is |
| 37 | $S_4\times C_2$ | \( 1 - 22 T + 251 T^{2} - 1836 T^{3} + 251 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.aw_jr_acsq |
| 41 | $S_4\times C_2$ | \( 1 + 11 T + 132 T^{2} + 828 T^{3} + 132 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.l_fc_bfw |
| 43 | $S_4\times C_2$ | \( 1 - 7 T + 106 T^{2} - 610 T^{3} + 106 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ah_ec_axm |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 125 T^{2} - 1000 T^{3} + 125 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.am_ev_abmm |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 104 T^{2} - 628 T^{3} + 104 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_ea_aye |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 97 T^{2} + 816 T^{3} + 97 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.i_dt_bfk |
| 61 | $S_4\times C_2$ | \( 1 - 27 T + 422 T^{2} - 3986 T^{3} + 422 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.abb_qg_afxi |
| 67 | $S_4\times C_2$ | \( 1 - 15 T + 228 T^{2} - 1794 T^{3} + 228 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ap_iu_acra |
| 71 | $S_4\times C_2$ | \( 1 - 2 T - 15 T^{2} + 564 T^{3} - 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ac_ap_vs |
| 73 | $S_4\times C_2$ | \( 1 + T + 68 T^{2} + 804 T^{3} + 68 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.73.b_cq_bey |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) | 3.79.m_kz_cxk |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 293 T^{2} + 2356 T^{3} + 293 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.o_lh_dmq |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 1580 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_ed_ciu |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 180 T^{2} - 740 T^{3} + 180 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.aj_gy_abcm |
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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
−6.71861174168812901647576859849, −6.42677664200808133019595086414, −6.41822014702298546737006540583, −6.40033220215857411425824997002, −5.81433488136102177958514243240, −5.80769598634998352812640450247, −5.78686367258602013074102082635, −5.37360625481194901224286903815, −5.27088768600363420127331681693, −5.10375648585788414404913174204, −4.53831378585146257757118163710, −4.50804974138129668742029522198, −4.02362040332324347804744828855, −3.80492250942555462135149779891, −3.76515129597664665357214246115, −3.10404342381267458433023239974, −2.70987575125322459964159157891, −2.69553688523424103159568618463, −2.47202387558783589583912477962, −2.12988041443074167509022562282, −1.77507553752995835058949062114, −1.42793343644474169913991114012, −0.835242867009325714520939706967, −0.68875027615424886941536215636, −0.43613490399659705165179248398,
0.43613490399659705165179248398, 0.68875027615424886941536215636, 0.835242867009325714520939706967, 1.42793343644474169913991114012, 1.77507553752995835058949062114, 2.12988041443074167509022562282, 2.47202387558783589583912477962, 2.69553688523424103159568618463, 2.70987575125322459964159157891, 3.10404342381267458433023239974, 3.76515129597664665357214246115, 3.80492250942555462135149779891, 4.02362040332324347804744828855, 4.50804974138129668742029522198, 4.53831378585146257757118163710, 5.10375648585788414404913174204, 5.27088768600363420127331681693, 5.37360625481194901224286903815, 5.78686367258602013074102082635, 5.80769598634998352812640450247, 5.81433488136102177958514243240, 6.40033220215857411425824997002, 6.41822014702298546737006540583, 6.42677664200808133019595086414, 6.71861174168812901647576859849
