| L(s) = 1 | − 3-s + 5-s − 3·7-s − 2·9-s − 3·11-s − 6·13-s − 15-s − 2·17-s − 4·19-s + 3·21-s − 9·23-s − 8·25-s + 3·27-s + 6·29-s − 7·31-s + 3·33-s − 3·35-s + 3·37-s + 6·39-s − 2·41-s − 2·45-s − 16·47-s + 6·49-s + 2·51-s − 6·53-s − 3·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.904·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.654·21-s − 1.87·23-s − 8/5·25-s + 0.577·27-s + 1.11·29-s − 1.25·31-s + 0.522·33-s − 0.507·35-s + 0.493·37-s + 0.960·39-s − 0.312·41-s − 0.298·45-s − 2.33·47-s + 6/7·49-s + 0.280·51-s − 0.824·53-s − 0.404·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_d_c |
| 5 | $S_4\times C_2$ | \( 1 - T + 9 T^{2} - 6 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ab_j_ag |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 124 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bj_eu |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 3 T^{2} - 60 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_d_aci |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 96 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_bp_ds |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 53 T^{2} + 206 T^{3} + 53 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.j_cb_hy |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 164 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_bj_agi |
| 31 | $S_4\times C_2$ | \( 1 + 7 T + 97 T^{2} + 406 T^{3} + 97 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.h_dt_pq |
| 37 | $S_4\times C_2$ | \( 1 - 3 T + 71 T^{2} - 2 p T^{3} + 71 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ad_ct_acw |
| 41 | $S_4\times C_2$ | \( 1 + 2 T - 5 T^{2} - 348 T^{3} - 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.c_af_ank |
| 43 | $S_4\times C_2$ | \( 1 + 65 T^{2} + 64 T^{3} + 65 p T^{4} + p^{3} T^{6} \) | 3.43.a_cn_cm |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 201 T^{2} + 1536 T^{3} + 201 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.q_ht_chc |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 452 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_ed_rk |
| 59 | $S_4\times C_2$ | \( 1 + 15 T + 195 T^{2} + 1558 T^{3} + 195 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.p_hn_chy |
| 61 | $S_4\times C_2$ | \( 1 - 9 T^{2} - 808 T^{3} - 9 p T^{4} + p^{3} T^{6} \) | 3.61.a_aj_abfc |
| 67 | $S_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 1222 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.aj_en_abva |
| 71 | $S_4\times C_2$ | \( 1 + 15 T + 185 T^{2} + 1346 T^{3} + 185 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.p_hd_bzu |
| 73 | $S_4\times C_2$ | \( 1 + 26 T + 419 T^{2} + 4212 T^{3} + 419 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ba_qd_gga |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 189 T^{2} + 188 T^{3} + 189 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.c_hh_hg |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 245 T^{2} + 964 T^{3} + 245 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.g_jl_blc |
| 89 | $S_4\times C_2$ | \( 1 - T + 115 T^{2} + 186 T^{3} + 115 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ab_el_he |
| 97 | $S_4\times C_2$ | \( 1 + 41 T + 839 T^{2} + 10350 T^{3} + 839 p T^{4} + 41 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.bp_bgh_pic |
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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
−9.320293235843839080979190998159, −8.737157524256351990717202688025, −8.343370751268794108510541498036, −8.267762065733110817106155065186, −7.897384860816542967392216514898, −7.87998066652821176460307796061, −7.18833575564830426839478036701, −7.16768037581756925167141811816, −6.72905795232243039201167932119, −6.58258673214190856142068528835, −6.09697677755765064698819873427, −5.89509649744003777659670106804, −5.82365509113599846190231723769, −5.33289558828902477452143967226, −5.27201962001146015901299075804, −4.60589748729596186412430845101, −4.49974913778059892117622075202, −4.03230455605509430165836704690, −3.97595656838871349472999057649, −3.07643636142947723321277634439, −2.90386512222548928668199798542, −2.85701940881975948177983525705, −2.24293463866612926046449195478, −1.76718589569726992088844392675, −1.59324165734509157696209295500, 0, 0, 0,
1.59324165734509157696209295500, 1.76718589569726992088844392675, 2.24293463866612926046449195478, 2.85701940881975948177983525705, 2.90386512222548928668199798542, 3.07643636142947723321277634439, 3.97595656838871349472999057649, 4.03230455605509430165836704690, 4.49974913778059892117622075202, 4.60589748729596186412430845101, 5.27201962001146015901299075804, 5.33289558828902477452143967226, 5.82365509113599846190231723769, 5.89509649744003777659670106804, 6.09697677755765064698819873427, 6.58258673214190856142068528835, 6.72905795232243039201167932119, 7.16768037581756925167141811816, 7.18833575564830426839478036701, 7.87998066652821176460307796061, 7.897384860816542967392216514898, 8.267762065733110817106155065186, 8.343370751268794108510541498036, 8.737157524256351990717202688025, 9.320293235843839080979190998159
