| L(s) = 1 | − 2·3-s − 3·5-s + 2·7-s − 9-s − 4·11-s − 6·13-s + 6·15-s − 8·19-s − 4·21-s − 2·23-s + 6·25-s + 4·27-s + 3·29-s − 4·31-s + 8·33-s − 6·35-s − 4·37-s + 12·39-s + 14·41-s − 18·43-s + 3·45-s + 2·47-s − 9·49-s − 6·53-s + 12·55-s + 16·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.755·7-s − 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.54·15-s − 1.83·19-s − 0.872·21-s − 0.417·23-s + 6/5·25-s + 0.769·27-s + 0.557·29-s − 0.718·31-s + 1.39·33-s − 1.01·35-s − 0.657·37-s + 1.92·39-s + 2.18·41-s − 2.74·43-s + 0.447·45-s + 0.291·47-s − 9/7·49-s − 0.824·53-s + 1.61·55-s + 2.11·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_i |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 32 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_n_abg |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 84 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_bh_dg |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 116 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bj_em |
| 17 | $S_4\times C_2$ | \( 1 + 23 T^{2} + 52 T^{3} + 23 p T^{4} + p^{3} T^{6} \) | 3.17.a_x_ca |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 69 T^{2} + 308 T^{3} + 69 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_cr_lw |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 88 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_cn_dk |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 65 T^{2} + 132 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.e_cn_fc |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 107 T^{2} + 276 T^{3} + 107 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.e_ed_kq |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ao_gt_abts |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 189 T^{2} + 1456 T^{3} + 189 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.s_hh_cea |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 304 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ac_cn_als |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 388 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_bz_oy |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 145 T^{2} + 672 T^{3} + 145 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.i_fp_zw |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 131 T^{2} + 684 T^{3} + 131 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.k_fb_bai |
| 67 | $S_4\times C_2$ | \( 1 + 26 T + 381 T^{2} + 56 p T^{3} + 381 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ba_or_foi |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 165 T^{2} + 648 T^{3} + 165 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_gj_yy |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 347 T^{2} + 3180 T^{3} + 347 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.u_nj_esi |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 1132 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.e_dp_bro |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 81 T^{2} + 96 T^{3} + 81 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.o_dd_ds |
| 89 | $S_4\times C_2$ | \( 1 - 22 T + 407 T^{2} - 4148 T^{3} + 407 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aw_pr_agdo |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 115 T^{2} - 172 T^{3} + 115 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ae_el_agq |
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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
−9.051152687093268761748124520537, −8.752577536541371190131994321339, −8.520639671294869211340759654291, −8.237084907361474488113598693044, −7.962273388700895410542777856644, −7.66969100124969520993081286884, −7.54798663079659319533512304545, −7.25699716096968861738755498117, −6.98593514770264025033792752164, −6.52323079437938361719919624975, −6.16983810774882504032387948551, −6.01581734009807422060920750782, −5.81345519715399259928444174118, −5.16991604627669649020987698237, −5.00422410675747259189941423415, −4.93240679859869155194793535534, −4.44525659583463715283573419450, −4.31193595951098153514019315954, −4.00752715486541866783501543151, −3.27566720763600159230373872435, −2.96102578644912834313670958732, −2.86580362066059098182257105810, −2.24726801042318342357932078609, −1.78836548999288821023160436933, −1.34860436452899189187683807873, 0, 0, 0,
1.34860436452899189187683807873, 1.78836548999288821023160436933, 2.24726801042318342357932078609, 2.86580362066059098182257105810, 2.96102578644912834313670958732, 3.27566720763600159230373872435, 4.00752715486541866783501543151, 4.31193595951098153514019315954, 4.44525659583463715283573419450, 4.93240679859869155194793535534, 5.00422410675747259189941423415, 5.16991604627669649020987698237, 5.81345519715399259928444174118, 6.01581734009807422060920750782, 6.16983810774882504032387948551, 6.52323079437938361719919624975, 6.98593514770264025033792752164, 7.25699716096968861738755498117, 7.54798663079659319533512304545, 7.66969100124969520993081286884, 7.962273388700895410542777856644, 8.237084907361474488113598693044, 8.520639671294869211340759654291, 8.752577536541371190131994321339, 9.051152687093268761748124520537
