| L(s) = 1 | + 2-s + 4·7-s − 8-s − 9·11-s − 6·13-s + 4·14-s − 3·16-s + 2·17-s − 4·19-s − 9·22-s + 23-s − 6·26-s + 3·29-s + 2·34-s − 13·37-s − 4·38-s − 13·41-s + 13·43-s + 46-s + 2·47-s + 2·49-s − 3·53-s − 4·56-s + 3·58-s − 22·59-s + 10·61-s − 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·7-s − 0.353·8-s − 2.71·11-s − 1.66·13-s + 1.06·14-s − 3/4·16-s + 0.485·17-s − 0.917·19-s − 1.91·22-s + 0.208·23-s − 1.17·26-s + 0.557·29-s + 0.342·34-s − 2.13·37-s − 0.648·38-s − 2.03·41-s + 1.98·43-s + 0.147·46-s + 0.291·47-s + 2/7·49-s − 0.412·53-s − 0.534·56-s + 0.393·58-s − 2.86·59-s + 1.28·61-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_b_a |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 2 p T^{2} - 6 p T^{3} + 2 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_o_abq |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) | 3.11.j_ci_ir |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 38 T^{2} + 154 T^{3} + 38 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bm_fy |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 40 T^{2} - 60 T^{3} + 40 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ac_bo_aci |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 64 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_bh_cm |
| 23 | $S_4\times C_2$ | \( 1 - T + 13 T^{2} + 66 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ab_n_co |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.31.a_dp_a |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 706 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.n_eh_bbe |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1094 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.n_gl_bqc |
| 43 | $S_4\times C_2$ | \( 1 - 13 T + 131 T^{2} - 810 T^{3} + 131 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.an_fb_abfe |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 72 T^{2} + 78 T^{3} + 72 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ac_cu_da |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 45 T^{2} + 634 T^{3} + 45 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.d_bt_yk |
| 59 | $S_4\times C_2$ | \( 1 + 22 T + 317 T^{2} + 2852 T^{3} + 317 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.w_mf_efs |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 187 T^{2} - 1108 T^{3} + 187 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ak_hf_abqq |
| 67 | $S_4\times C_2$ | \( 1 - 28 T + 406 T^{2} - 3946 T^{3} + 406 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.abc_pq_afvu |
| 71 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 488 T^{3} + 21 p T^{4} + p^{3} T^{6} \) | 3.71.a_v_asu |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 35 T^{2} - 10 p T^{3} + 35 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.d_bj_abcc |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 97 T^{2} + 92 T^{3} + 97 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.c_dt_do |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 311 T^{2} + 2534 T^{3} + 311 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.p_lz_dtm |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 450 T^{2} + 4738 T^{3} + 450 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.be_ri_hag |
| 97 | $S_4\times C_2$ | \( 1 - T + 149 T^{2} - 270 T^{3} + 149 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ab_ft_akk |
| 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
−7.64302673146681872008987925823, −7.11454644196469675432859931926, −6.90084854731077425834493180007, −6.76373011611647281411267704386, −6.70552802867452933505476504493, −6.03519209783934509962549554488, −6.01691498872947156790231046405, −5.39848257259793696382675504556, −5.39590384328557906843515598190, −5.34403643744503218957345835443, −4.97919507843493303142217157430, −4.95009326479529262835584847000, −4.59951400712065440367328168593, −4.37141284972273576193065710639, −4.11363963947173755897773569535, −3.90870552729687028089328655057, −3.27537603319859411936327541447, −3.26535314332877966635901021664, −2.70331110199360862416504296533, −2.69606856536708854637336530511, −2.42562877559978535394363053453, −1.97633602874157303797320759700, −1.96077123243994726948658492965, −1.24942246275856124121968844579, −1.15604607510574457959240121820, 0, 0, 0,
1.15604607510574457959240121820, 1.24942246275856124121968844579, 1.96077123243994726948658492965, 1.97633602874157303797320759700, 2.42562877559978535394363053453, 2.69606856536708854637336530511, 2.70331110199360862416504296533, 3.26535314332877966635901021664, 3.27537603319859411936327541447, 3.90870552729687028089328655057, 4.11363963947173755897773569535, 4.37141284972273576193065710639, 4.59951400712065440367328168593, 4.95009326479529262835584847000, 4.97919507843493303142217157430, 5.34403643744503218957345835443, 5.39590384328557906843515598190, 5.39848257259793696382675504556, 6.01691498872947156790231046405, 6.03519209783934509962549554488, 6.70552802867452933505476504493, 6.76373011611647281411267704386, 6.90084854731077425834493180007, 7.11454644196469675432859931926, 7.64302673146681872008987925823
