| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 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.f_s_bq |
| 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.c_p_u |
| 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.e_bx_gm |
| 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.k_dv_tw |
| 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.ah_ca_ais |
| 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.w_jr_csq |
| 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.m_ev_bmm |
| 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.j_ea_ye |
| 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.bb_qg_fxi |
| 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.c_ap_avs |
| 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.am_kz_acxk |
| 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 |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.433568853606045203390414738843, −8.746939532838198898775724779684, −8.741982756997234102511269938551, −8.573531536505593772630682869378, −8.034967079381806913777910369105, −7.988369776839222903294363001565, −7.64614582058964897520393043134, −7.46347984706838027694623059555, −7.01190092877990980344966375775, −6.94310935114716150109795689340, −6.23345326727664724435997948950, −6.15904428139229912113815858097, −6.08766297204798646215574776909, −5.24855070407140499559819472837, −5.18491043485289062408372557616, −5.12182810815271383672389337463, −4.54742915557564384178882243886, −4.42278487397786418165709040523, −3.97490405637998675436297106394, −3.60852019758642547597988303410, −3.33799663443975421161782242422, −2.98226586704130845961617796600, −2.02068481639020879856745582001, −1.96564889349857540777405033955, −1.31958477614384609273745806622, 0, 0, 0,
1.31958477614384609273745806622, 1.96564889349857540777405033955, 2.02068481639020879856745582001, 2.98226586704130845961617796600, 3.33799663443975421161782242422, 3.60852019758642547597988303410, 3.97490405637998675436297106394, 4.42278487397786418165709040523, 4.54742915557564384178882243886, 5.12182810815271383672389337463, 5.18491043485289062408372557616, 5.24855070407140499559819472837, 6.08766297204798646215574776909, 6.15904428139229912113815858097, 6.23345326727664724435997948950, 6.94310935114716150109795689340, 7.01190092877990980344966375775, 7.46347984706838027694623059555, 7.64614582058964897520393043134, 7.988369776839222903294363001565, 8.034967079381806913777910369105, 8.573531536505593772630682869378, 8.741982756997234102511269938551, 8.746939532838198898775724779684, 9.433568853606045203390414738843
