| L(s) = 1 | − 2·4-s − 4·7-s + 2·8-s + 7·11-s − 7·17-s − 11·19-s − 5·23-s + 8·28-s + 5·29-s − 3·31-s − 8·32-s − 26·37-s + 8·41-s − 8·43-s − 14·44-s + 10·47-s − 5·49-s − 11·53-s − 8·56-s + 14·59-s + 12·61-s + 4·64-s − 3·67-s + 14·68-s + 26·71-s − 12·73-s + 22·76-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 0.707·8-s + 2.11·11-s − 1.69·17-s − 2.52·19-s − 1.04·23-s + 1.51·28-s + 0.928·29-s − 0.538·31-s − 1.41·32-s − 4.27·37-s + 1.24·41-s − 1.21·43-s − 2.11·44-s + 1.45·47-s − 5/7·49-s − 1.51·53-s − 1.06·56-s + 1.82·59-s + 1.53·61-s + 1/2·64-s − 0.366·67-s + 1.69·68-s + 3.08·71-s − 1.40·73-s + 2.52·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} - p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) | 3.2.a_c_ac |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 52 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_v_ca |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 36 T^{2} - 117 T^{3} + 36 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ah_bk_aen |
| 13 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 62 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.13.a_f_ack |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 44 T^{2} + 179 T^{3} + 44 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.h_bs_gx |
| 19 | $S_4\times C_2$ | \( 1 + 11 T + 84 T^{2} + 419 T^{3} + 84 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.l_dg_qd |
| 23 | $S_4\times C_2$ | \( 1 + 5 T + 12 T^{2} - 9 T^{3} + 12 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.f_m_aj |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 92 T^{2} - 289 T^{3} + 92 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.af_do_ald |
| 37 | $S_4\times C_2$ | \( 1 + 26 T + 9 p T^{2} + 2546 T^{3} + 9 p^{2} T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ba_mv_dty |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 107 T^{2} - 526 T^{3} + 107 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ai_ed_aug |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 57 T^{2} + 320 T^{3} + 57 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.i_cf_mi |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 121 T^{2} - 948 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ak_er_abkm |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 120 T^{2} + 1191 T^{3} + 120 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.l_eq_btv |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 181 T^{2} - 1350 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ao_gz_abzy |
| 61 | $S_4\times C_2$ | \( 1 - 12 T + 167 T^{2} - 1400 T^{3} + 167 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.am_gl_acbw |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 44 T^{2} + 671 T^{3} + 44 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.d_bs_zv |
| 71 | $S_4\times C_2$ | \( 1 - 26 T + 401 T^{2} - 3940 T^{3} + 401 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aba_pl_afvo |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 251 T^{2} + 1768 T^{3} + 251 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.m_jr_cqa |
| 79 | $S_4\times C_2$ | \( 1 - 10 T + 255 T^{2} - 1546 T^{3} + 255 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ak_jv_achm |
| 83 | $S_4\times C_2$ | \( 1 + 5 T + 208 T^{2} + 651 T^{3} + 208 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.f_ia_zb |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 102 T^{2} - 79 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aj_dy_adb |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 114 T^{2} - 1529 T^{3} + 114 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.af_ek_acgv |
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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
−7.17078238487953464802892689033, −7.10730485608184475359562226660, −6.74330396197272121037271470402, −6.62231512788487208306731639060, −6.43094850572234982010121189739, −6.35719696215255263520131377460, −6.27988265327913343313190922584, −5.64920222627072130411806599933, −5.54756519287934772306552593218, −5.18667298546388416813532504153, −4.76079672023662800073139941235, −4.67574232913596602907931002610, −4.63719697371629761980405210788, −4.00072468587682674198592457295, −3.97736980745917715450716997413, −3.85192186316182830073683047689, −3.55194866955826330703100175865, −3.35499738909050301764921963201, −3.06124995330286252142169159762, −2.25924458085128941666757207611, −2.23583722186031707231218993164, −1.94514770929757642862476233357, −1.92589493521627963105639112957, −1.12946884200180039229664387211, −0.984899974874516840922938564223, 0, 0, 0,
0.984899974874516840922938564223, 1.12946884200180039229664387211, 1.92589493521627963105639112957, 1.94514770929757642862476233357, 2.23583722186031707231218993164, 2.25924458085128941666757207611, 3.06124995330286252142169159762, 3.35499738909050301764921963201, 3.55194866955826330703100175865, 3.85192186316182830073683047689, 3.97736980745917715450716997413, 4.00072468587682674198592457295, 4.63719697371629761980405210788, 4.67574232913596602907931002610, 4.76079672023662800073139941235, 5.18667298546388416813532504153, 5.54756519287934772306552593218, 5.64920222627072130411806599933, 6.27988265327913343313190922584, 6.35719696215255263520131377460, 6.43094850572234982010121189739, 6.62231512788487208306731639060, 6.74330396197272121037271470402, 7.10730485608184475359562226660, 7.17078238487953464802892689033
