| L(s) = 1 | − 2·2-s + 3·4-s − 4·7-s − 4·8-s − 11-s − 9·13-s + 8·14-s + 3·16-s + 17-s − 4·19-s + 2·22-s + 11·23-s + 18·26-s − 12·28-s − 2·29-s − 5·31-s − 6·32-s − 2·34-s + 8·38-s + 15·41-s + 3·43-s − 3·44-s − 22·46-s − 2·47-s − 2·49-s − 27·52-s − 5·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s − 0.301·11-s − 2.49·13-s + 2.13·14-s + 3/4·16-s + 0.242·17-s − 0.917·19-s + 0.426·22-s + 2.29·23-s + 3.53·26-s − 2.26·28-s − 0.371·29-s − 0.898·31-s − 1.06·32-s − 0.342·34-s + 1.29·38-s + 2.34·41-s + 0.457·43-s − 0.452·44-s − 3.24·46-s − 0.291·47-s − 2/7·49-s − 3.74·52-s − 0.686·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 43^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 43^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 43 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_b_a |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 18 T^{2} + 46 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_s_bu |
| 11 | $S_4\times C_2$ | \( 1 + T + 14 T^{2} + 47 T^{3} + 14 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.11.b_o_bv |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) | 3.13.j_co_kb |
| 17 | $S_4\times C_2$ | \( 1 - T + 43 T^{2} - 30 T^{3} + 43 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ab_br_abe |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 2 p T^{2} + 150 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_bm_fu |
| 23 | $S_4\times C_2$ | \( 1 - 11 T + 37 T^{2} - 54 T^{3} + 37 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.al_bl_acc |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 82 T^{2} + 108 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_de_ee |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 77 T^{2} + 246 T^{3} + 77 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.f_cz_jm |
| 37 | $S_4\times C_2$ | \( 1 + 71 T^{2} - 64 T^{3} + 71 p T^{4} + p^{3} T^{6} \) | 3.37.a_ct_acm |
| 41 | $S_4\times C_2$ | \( 1 - 15 T + 155 T^{2} - 1198 T^{3} + 155 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ap_fz_abuc |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 8 T^{2} - 476 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_i_asi |
| 53 | $S_4\times C_2$ | \( 1 + 5 T + 143 T^{2} + 466 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.f_fn_ry |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 165 T^{2} + 864 T^{3} + 165 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.i_gj_bhg |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 191 T^{2} + 1440 T^{3} + 191 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.q_hj_cdk |
| 67 | $S_4\times C_2$ | \( 1 - 11 T + 121 T^{2} - 1142 T^{3} + 121 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.al_er_abry |
| 71 | $S_4\times C_2$ | \( 1 + 22 T + 297 T^{2} + 2700 T^{3} + 297 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.w_ll_dzw |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 271 T^{2} - 2320 T^{3} + 271 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aq_kl_adlg |
| 79 | $S_4\times C_2$ | \( 1 - 24 T + 389 T^{2} - 4048 T^{3} + 389 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ay_oz_afzs |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 170 T^{2} + 677 T^{3} + 170 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.h_go_bab |
| 89 | $S_4\times C_2$ | \( 1 - 38 T + 723 T^{2} - 8508 T^{3} + 723 p T^{4} - 38 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.abm_bbv_ampg |
| 97 | $S_4\times C_2$ | \( 1 + T + 214 T^{2} - 83 T^{3} + 214 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.97.b_ig_adf |
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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
−7.10074224122704070118654760569, −6.86132986652562306956705451316, −6.82932042503882766674415601579, −6.59659658466015305581701531681, −6.30429013689429958019034714852, −6.06470790640337593356752631267, −5.97905327903147011534681363979, −5.46885347516392311493639927490, −5.36763758180777455890692942694, −5.02338782813244348499907271989, −4.85576216882245095265376394924, −4.65607586356407444508632203829, −4.41419160186490764630939119953, −3.90253748021036350309402465033, −3.78170832289246370215399466452, −3.37201100640860692261030623106, −3.12883957284444333850250360483, −2.98834700022929616800229546205, −2.60198609631381282735163907297, −2.49903337048280855741765925635, −2.10784763786085020519753711792, −2.01918716772857776788422049166, −1.52438668648724392382700571569, −0.953253850716875371532460202536, −0.896912986169064803367874093887, 0, 0, 0,
0.896912986169064803367874093887, 0.953253850716875371532460202536, 1.52438668648724392382700571569, 2.01918716772857776788422049166, 2.10784763786085020519753711792, 2.49903337048280855741765925635, 2.60198609631381282735163907297, 2.98834700022929616800229546205, 3.12883957284444333850250360483, 3.37201100640860692261030623106, 3.78170832289246370215399466452, 3.90253748021036350309402465033, 4.41419160186490764630939119953, 4.65607586356407444508632203829, 4.85576216882245095265376394924, 5.02338782813244348499907271989, 5.36763758180777455890692942694, 5.46885347516392311493639927490, 5.97905327903147011534681363979, 6.06470790640337593356752631267, 6.30429013689429958019034714852, 6.59659658466015305581701531681, 6.82932042503882766674415601579, 6.86132986652562306956705451316, 7.10074224122704070118654760569
