| L(s) = 1 | − 2·5-s − 3·7-s + 4·11-s − 3·13-s − 3·17-s − 5·19-s + 7·23-s − 3·25-s − 4·29-s − 7·31-s + 6·35-s − 10·37-s + 10·41-s − 9·43-s + 47-s − 15·49-s − 22·53-s − 8·55-s + 20·59-s − 2·61-s + 6·65-s − 13·67-s − 20·73-s − 12·77-s − 7·79-s + 83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s + 1.20·11-s − 0.832·13-s − 0.727·17-s − 1.14·19-s + 1.45·23-s − 3/5·25-s − 0.742·29-s − 1.25·31-s + 1.01·35-s − 1.64·37-s + 1.56·41-s − 1.37·43-s + 0.145·47-s − 2.14·49-s − 3.02·53-s − 1.07·55-s + 2.60·59-s − 0.256·61-s + 0.744·65-s − 1.58·67-s − 2.34·73-s − 1.36·77-s − 0.787·79-s + 0.109·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 23 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_h_x |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) | 3.7.d_y_br |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 57 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ae_x_acf |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 30 T^{2} + 120 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.d_be_eq |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 56 T^{2} + 188 T^{3} + 56 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.f_ce_hg |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 26 T^{2} - 30 T^{3} + 26 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ah_ba_abe |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 36 T^{2} + 214 T^{3} + 36 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.e_bk_ig |
| 31 | $S_4\times C_2$ | \( 1 + 7 T + 5 T^{2} - 194 T^{3} + 5 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.h_f_ahm |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 134 T^{2} + 742 T^{3} + 134 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.k_fe_bco |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 52 T^{2} - 112 T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ak_ca_aei |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 114 T^{2} + 778 T^{3} + 114 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.j_ek_bdy |
| 47 | $S_4\times C_2$ | \( 1 - T + 60 T^{2} + 4 p T^{3} + 60 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ab_ci_hg |
| 53 | $S_4\times C_2$ | \( 1 + 22 T + 261 T^{2} + 2125 T^{3} + 261 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.w_kb_ddt |
| 59 | $S_4\times C_2$ | \( 1 - 20 T + 300 T^{2} - 2588 T^{3} + 300 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.au_lo_advo |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 174 T^{2} + 238 T^{3} + 174 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_gs_je |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 168 T^{2} + 1694 T^{3} + 168 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.n_gm_cne |
| 71 | $S_4\times C_2$ | \( 1 + 45 T^{2} + 824 T^{3} + 45 p T^{4} + p^{3} T^{6} \) | 3.71.a_bt_bfs |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 293 T^{2} + 2987 T^{3} + 293 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.u_lh_ekx |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 102 T^{2} + 1202 T^{3} + 102 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.h_dy_bug |
| 83 | $S_4\times C_2$ | \( 1 - T + 100 T^{2} + 455 T^{3} + 100 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ab_dw_rn |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 271 T^{2} + 2204 T^{3} + 271 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.o_kl_dgu |
| 97 | $S_4\times C_2$ | \( 1 + 7 T + 251 T^{2} + 1294 T^{3} + 251 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.h_jr_bxu |
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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
−8.402923052457489041762098985810, −7.68071968166602800510516097980, −7.67609985193935387231207242174, −7.54839965978314330449005505797, −6.90244871936415316538888726171, −6.88683251894850487208421277021, −6.87815054958282834041976947308, −6.44181675718703235527711560433, −6.17286713701210335779387306601, −6.02528643280836521826710835133, −5.56699941341924489575235551516, −5.22341129077437813486530201916, −5.11526773408566497527734576242, −4.67062636591440331454672833698, −4.33836754351832949105727169265, −4.26867759931535712608984197852, −3.72499412150607192256713744129, −3.69775183459118376624871317664, −3.39128214972577937820278759831, −2.92845528955063900513531478208, −2.74762703123033452635943006671, −2.39898517173195783368675435121, −1.73992260968178082565115147835, −1.54020445761742362092570963844, −1.28214497025783749742719978427, 0, 0, 0,
1.28214497025783749742719978427, 1.54020445761742362092570963844, 1.73992260968178082565115147835, 2.39898517173195783368675435121, 2.74762703123033452635943006671, 2.92845528955063900513531478208, 3.39128214972577937820278759831, 3.69775183459118376624871317664, 3.72499412150607192256713744129, 4.26867759931535712608984197852, 4.33836754351832949105727169265, 4.67062636591440331454672833698, 5.11526773408566497527734576242, 5.22341129077437813486530201916, 5.56699941341924489575235551516, 6.02528643280836521826710835133, 6.17286713701210335779387306601, 6.44181675718703235527711560433, 6.87815054958282834041976947308, 6.88683251894850487208421277021, 6.90244871936415316538888726171, 7.54839965978314330449005505797, 7.67609985193935387231207242174, 7.68071968166602800510516097980, 8.402923052457489041762098985810
