| L(s) = 1 | − 2·5-s + 4·7-s − 6·13-s − 2·17-s − 10·19-s − 12·23-s − 3·25-s − 6·29-s + 14·31-s − 8·35-s − 3·37-s − 20·41-s + 2·43-s − 8·47-s + 3·49-s − 4·59-s − 18·61-s + 12·65-s + 4·67-s − 16·71-s − 6·73-s + 14·79-s + 4·85-s − 10·89-s − 24·91-s + 20·95-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.66·13-s − 0.485·17-s − 2.29·19-s − 2.50·23-s − 3/5·25-s − 1.11·29-s + 2.51·31-s − 1.35·35-s − 0.493·37-s − 3.12·41-s + 0.304·43-s − 1.16·47-s + 3/7·49-s − 0.520·59-s − 2.30·61-s + 1.48·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s + 1.57·79-s + 0.433·85-s − 1.05·89-s − 2.51·91-s + 2.05·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 24 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_h_y |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 40 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_n_abo |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.11.a_bh_a |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.13.g_bz_gi |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 120 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_t_eq |
| 19 | $S_4\times C_2$ | \( 1 + 10 T + 77 T^{2} + 372 T^{3} + 77 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.k_cz_oi |
| 23 | $S_4\times C_2$ | \( 1 + 12 T + 89 T^{2} + 452 T^{3} + 89 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.m_dl_rk |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 71 T^{2} + 352 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_ct_no |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 97 T^{2} - 524 T^{3} + 97 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ao_dt_aue |
| 41 | $S_4\times C_2$ | \( 1 + 20 T + 235 T^{2} + 1800 T^{3} + 235 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.u_jb_crg |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 117 T^{2} - 164 T^{3} + 117 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_en_agi |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 125 T^{2} + 592 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.i_ev_wu |
| 53 | $S_4\times C_2$ | \( 1 + 95 T^{2} + 128 T^{3} + 95 p T^{4} + p^{3} T^{6} \) | 3.53.a_dr_ey |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 149 T^{2} + 356 T^{3} + 149 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.e_ft_ns |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 227 T^{2} + 1900 T^{3} + 227 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.s_it_cvc |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 145 T^{2} - 616 T^{3} + 145 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ae_fp_axs |
| 71 | $S_4\times C_2$ | \( 1 + 16 T + 261 T^{2} + 2304 T^{3} + 261 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.q_kb_dkq |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 164 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_dj_gi |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 209 T^{2} - 1740 T^{3} + 209 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ao_ib_acoy |
| 83 | $S_4\times C_2$ | \( 1 + 137 T^{2} + 416 T^{3} + 137 p T^{4} + p^{3} T^{6} \) | 3.83.a_fh_qa |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 291 T^{2} + 1776 T^{3} + 291 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.k_lf_cqi |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 239 T^{2} + 916 T^{3} + 239 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_jf_bjg |
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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
−8.331615695016953087719704739626, −7.86624576129462334843293130546, −7.81581146234093514474338033562, −7.67576507108907682023867853879, −7.31060630817216553421987775531, −6.92608374405110211061917337065, −6.75873258903580878046038119488, −6.29057358737739135644832619515, −6.21691935919021714227159383821, −6.15524542876267894207265007538, −5.45876977341914068367625829597, −5.29773835687243330138674612690, −5.01822713409946904922372278556, −4.58625102369225993754074025050, −4.52733934308479206009365991413, −4.42153782586921480383820080959, −3.94477863471075810375596354968, −3.72238155692514267644765265764, −3.48573669565879844455356259956, −2.75592851265773419465711582556, −2.65194811827012881805296121403, −2.28477246985030943542578132500, −1.86664963773763607789367829252, −1.55155617902389377613318688746, −1.45452163665262530215293038588, 0, 0, 0,
1.45452163665262530215293038588, 1.55155617902389377613318688746, 1.86664963773763607789367829252, 2.28477246985030943542578132500, 2.65194811827012881805296121403, 2.75592851265773419465711582556, 3.48573669565879844455356259956, 3.72238155692514267644765265764, 3.94477863471075810375596354968, 4.42153782586921480383820080959, 4.52733934308479206009365991413, 4.58625102369225993754074025050, 5.01822713409946904922372278556, 5.29773835687243330138674612690, 5.45876977341914068367625829597, 6.15524542876267894207265007538, 6.21691935919021714227159383821, 6.29057358737739135644832619515, 6.75873258903580878046038119488, 6.92608374405110211061917337065, 7.31060630817216553421987775531, 7.67576507108907682023867853879, 7.81581146234093514474338033562, 7.86624576129462334843293130546, 8.331615695016953087719704739626
