| L(s) = 1 | + 3·3-s + 2·5-s + 6·9-s − 5·11-s + 2·13-s + 6·15-s − 7·17-s − 4·19-s − 4·23-s − 9·25-s + 10·27-s − 29-s − 5·31-s − 15·33-s + 37-s + 6·39-s + 3·41-s − 23·43-s + 12·45-s − 7·47-s − 17·49-s − 21·51-s − 10·55-s − 12·57-s − 6·59-s + 33·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 2·9-s − 1.50·11-s + 0.554·13-s + 1.54·15-s − 1.69·17-s − 0.917·19-s − 0.834·23-s − 9/5·25-s + 1.92·27-s − 0.185·29-s − 0.898·31-s − 2.61·33-s + 0.164·37-s + 0.960·39-s + 0.468·41-s − 3.50·43-s + 1.78·45-s − 1.02·47-s − 2.42·49-s − 2.94·51-s − 1.34·55-s − 1.58·57-s − 0.781·59-s + 4.22·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 18 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_n_as |
| 7 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 2 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.7.a_r_c |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 28 T^{2} + 111 T^{3} + 28 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.f_bc_eh |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 2 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_t_ac |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 46 T^{2} + 237 T^{3} + 46 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.h_bu_jd |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 154 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_z_fy |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_p_aco |
| 29 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + 21 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.29.b_e_v |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 64 T^{2} + 173 T^{3} + 64 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.f_cm_gr |
| 37 | $S_4\times C_2$ | \( 1 - T + 106 T^{2} - 73 T^{3} + 106 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ab_ec_acv |
| 43 | $S_4\times C_2$ | \( 1 + 23 T + 300 T^{2} + 2387 T^{3} + 300 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.x_lo_dnv |
| 47 | $S_4\times C_2$ | \( 1 + 7 T + 142 T^{2} + 615 T^{3} + 142 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.h_fm_xr |
| 53 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 20 T^{3} + 111 p T^{4} + p^{3} T^{6} \) | 3.53.a_eh_u |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 173 T^{2} + 700 T^{3} + 173 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.g_gr_bay |
| 61 | $S_4\times C_2$ | \( 1 - 33 T + 530 T^{2} - 5165 T^{3} + 530 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.abh_uk_ahqr |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 281 T^{2} + 2616 T^{3} + 281 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.u_kv_dwq |
| 71 | $S_4\times C_2$ | \( 1 + 17 T + 158 T^{2} + 1141 T^{3} + 158 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.r_gc_brx |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 230 T^{2} + 1277 T^{3} + 230 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.j_iw_bxd |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 229 T^{2} - 2320 T^{3} + 229 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aq_iv_adlg |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 219 T^{2} + 1838 T^{3} + 219 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.s_il_css |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 311 T^{2} + 3164 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.s_lz_ers |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 231 T^{2} + 826 T^{3} + 231 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_ix_bfu |
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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
−7.42515574214827479483017609649, −6.96879620821970174962512743899, −6.81819968149263237404801348627, −6.77564726617198021679534760962, −6.24205471502661034779863480453, −6.16331851949778794016753791953, −6.08122208247510886275562956562, −5.51514243911407929043752935365, −5.36536392753328367421998870368, −5.35271677546823239568863564161, −4.65835785239273559368132684133, −4.65368083559456026420222994361, −4.53315617149731864012683384400, −4.04207286327161926637425413549, −3.77604596828720205683016961125, −3.77355945946450781476150677746, −3.17390947975819621978992921209, −3.14678796639332760003547723409, −2.86227121656244367237761635061, −2.35658568108517876894036823374, −2.25985442738784463656019493106, −2.04539903002997045928366731640, −1.77693384087115765648481263560, −1.43150482564265508167086746501, −1.32714410770698432858217684703, 0, 0, 0,
1.32714410770698432858217684703, 1.43150482564265508167086746501, 1.77693384087115765648481263560, 2.04539903002997045928366731640, 2.25985442738784463656019493106, 2.35658568108517876894036823374, 2.86227121656244367237761635061, 3.14678796639332760003547723409, 3.17390947975819621978992921209, 3.77355945946450781476150677746, 3.77604596828720205683016961125, 4.04207286327161926637425413549, 4.53315617149731864012683384400, 4.65368083559456026420222994361, 4.65835785239273559368132684133, 5.35271677546823239568863564161, 5.36536392753328367421998870368, 5.51514243911407929043752935365, 6.08122208247510886275562956562, 6.16331851949778794016753791953, 6.24205471502661034779863480453, 6.77564726617198021679534760962, 6.81819968149263237404801348627, 6.96879620821970174962512743899, 7.42515574214827479483017609649
