| L(s) = 1 | + 2·3-s − 2·7-s − 9-s − 4·11-s + 6·13-s − 8·19-s − 4·21-s + 2·23-s − 4·27-s + 3·29-s − 4·31-s − 8·33-s + 4·37-s + 12·39-s + 14·41-s + 18·43-s − 2·47-s − 9·49-s + 6·53-s − 16·57-s − 8·59-s − 10·61-s + 2·63-s + 26·67-s + 4·69-s − 4·71-s + 20·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s − 1/3·9-s − 1.20·11-s + 1.66·13-s − 1.83·19-s − 0.872·21-s + 0.417·23-s − 0.769·27-s + 0.557·29-s − 0.718·31-s − 1.39·33-s + 0.657·37-s + 1.92·39-s + 2.18·41-s + 2.74·43-s − 0.291·47-s − 9/7·49-s + 0.824·53-s − 2.11·57-s − 1.04·59-s − 1.28·61-s + 0.251·63-s + 3.17·67-s + 0.481·69-s − 0.474·71-s + 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.622567789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.622567789\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 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.ac_f_ai |
| 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.c_n_bg |
| 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.ag_bj_aem |
| 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_aca |
| 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.ac_cn_adk |
| 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.ae_ed_akq |
| 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.as_hh_acea |
| 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.c_cn_ls |
| 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.ag_bz_aoy |
| 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.aba_or_afoi |
| 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.au_nj_aesi |
| 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.ao_dd_ads |
| 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.e_el_gq |
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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
−7.32518373139518129932743847594, −6.91400647340674355972332874351, −6.59039657156940479621702990776, −6.39081421273212804970106033659, −6.12067714675722880024549564354, −6.07684867685803144114507208394, −5.99436828420265932627666033465, −5.45889891843528060525136875486, −5.25957019217851972017510706995, −4.90158723438524163152522226910, −4.75970335126837053149331970194, −4.38241663293421317983369878591, −4.11303815967464856815877719079, −3.67699491154203262010126638947, −3.63626122238416930930356375149, −3.56470687722616393654532493944, −2.87866180242519779439723758909, −2.79929170569789157885945234973, −2.65091895868458916114258641261, −2.17806929521485719404300817977, −2.13700673107348871544623810274, −1.68430843600145333123485964547, −1.06017737021434199453814467594, −0.58408975449964908065751933249, −0.53944618493662514556937772009,
0.53944618493662514556937772009, 0.58408975449964908065751933249, 1.06017737021434199453814467594, 1.68430843600145333123485964547, 2.13700673107348871544623810274, 2.17806929521485719404300817977, 2.65091895868458916114258641261, 2.79929170569789157885945234973, 2.87866180242519779439723758909, 3.56470687722616393654532493944, 3.63626122238416930930356375149, 3.67699491154203262010126638947, 4.11303815967464856815877719079, 4.38241663293421317983369878591, 4.75970335126837053149331970194, 4.90158723438524163152522226910, 5.25957019217851972017510706995, 5.45889891843528060525136875486, 5.99436828420265932627666033465, 6.07684867685803144114507208394, 6.12067714675722880024549564354, 6.39081421273212804970106033659, 6.59039657156940479621702990776, 6.91400647340674355972332874351, 7.32518373139518129932743847594
