| L(s) = 1 | − 2·2-s − 8·7-s + 2·8-s + 3·11-s − 4·13-s + 16·14-s − 7·17-s + 9·19-s − 6·22-s − 9·23-s + 8·26-s + 9·29-s + 3·31-s + 14·34-s − 8·37-s − 18·38-s + 2·41-s − 4·43-s + 18·46-s + 20·47-s + 27·49-s + 25·53-s − 16·56-s − 18·58-s + 2·59-s + 2·61-s − 6·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3.02·7-s + 0.707·8-s + 0.904·11-s − 1.10·13-s + 4.27·14-s − 1.69·17-s + 2.06·19-s − 1.27·22-s − 1.87·23-s + 1.56·26-s + 1.67·29-s + 0.538·31-s + 2.40·34-s − 1.31·37-s − 2.91·38-s + 0.312·41-s − 0.609·43-s + 2.65·46-s + 2.91·47-s + 27/7·49-s + 3.43·53-s − 2.13·56-s − 2.36·58-s + 0.260·59-s + 0.256·61-s − 0.762·62-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 | $S_4\times C_2$ | \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + p^{3} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_e_g |
| 7 | $S_4\times C_2$ | \( 1 + 8 T + 37 T^{2} + 116 T^{3} + 37 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.i_bl_em |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 32 T^{2} - 65 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ad_bg_acn |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 114 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.e_x_ek |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) | 3.19.aj_dg_aof |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 4 p T^{2} + 431 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.j_do_qp |
| 29 | $S_4\times C_2$ | \( 1 - 9 T + 80 T^{2} - 509 T^{3} + 80 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.aj_dc_atp |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 119 T^{2} + 594 T^{3} + 119 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_ep_ww |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 298 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_cn_alm |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 280 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_bx_ku |
| 47 | $S_4\times C_2$ | \( 1 - 20 T + 261 T^{2} - 2088 T^{3} + 261 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.au_kb_adci |
| 53 | $S_4\times C_2$ | \( 1 - 25 T + 288 T^{2} - 2301 T^{3} + 288 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.az_lc_adkn |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 162 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ac_dz_agg |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ac_fb_ahw |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 236 T^{2} + 2011 T^{3} + 236 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.p_jc_czj |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 101 T^{2} + 520 T^{3} + 101 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.m_dx_ua |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 71 T^{2} + 588 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.c_ct_wq |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1982 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.am_gj_acyg |
| 83 | $S_4\times C_2$ | \( 1 + 13 T + 192 T^{2} + 1287 T^{3} + 192 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.n_hk_bxn |
| 89 | $S_4\times C_2$ | \( 1 + 15 T + 338 T^{2} + 2773 T^{3} + 338 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.p_na_ecr |
| 97 | $S_4\times C_2$ | \( 1 + 7 T + 262 T^{2} + 1255 T^{3} + 262 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.h_kc_bwh |
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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
−7.41851467524818787739994097884, −7.18005014779322959888260283558, −7.10275674714411160090965968612, −6.60736938090572688280207777532, −6.53990033046792016285008051359, −6.28015763317819638869172978197, −5.99300811871781195143689308922, −5.91838812726611440861010091046, −5.70437954165121241327268193124, −5.25964019809501516870195114975, −4.92796636840005887975237429752, −4.80976486230793698535133709427, −4.39126688501447952312176769629, −4.09502241006624557611857846530, −3.98670498258579528766396980018, −3.65255865840133938632704726184, −3.34506204885810724904478227889, −3.24923533117135754815551102809, −2.75467732264620662244030856090, −2.56218003353601339035917740097, −2.25939636565397108095161824300, −2.18297785428380382051913503801, −1.32135640832396013377385105055, −0.952143551060639675623784722284, −0.911912169435765673111129315948, 0, 0, 0,
0.911912169435765673111129315948, 0.952143551060639675623784722284, 1.32135640832396013377385105055, 2.18297785428380382051913503801, 2.25939636565397108095161824300, 2.56218003353601339035917740097, 2.75467732264620662244030856090, 3.24923533117135754815551102809, 3.34506204885810724904478227889, 3.65255865840133938632704726184, 3.98670498258579528766396980018, 4.09502241006624557611857846530, 4.39126688501447952312176769629, 4.80976486230793698535133709427, 4.92796636840005887975237429752, 5.25964019809501516870195114975, 5.70437954165121241327268193124, 5.91838812726611440861010091046, 5.99300811871781195143689308922, 6.28015763317819638869172978197, 6.53990033046792016285008051359, 6.60736938090572688280207777532, 7.10275674714411160090965968612, 7.18005014779322959888260283558, 7.41851467524818787739994097884
