| L(s) = 1 | − 3·2-s + 3-s + 6·4-s − 3·6-s − 2·7-s − 10·8-s − 6·9-s − 13·11-s + 6·12-s + 6·14-s + 15·16-s + 17-s + 18·18-s − 3·19-s − 2·21-s + 39·22-s + 10·23-s − 10·24-s − 8·27-s − 12·28-s − 8·29-s − 4·31-s − 21·32-s − 13·33-s − 3·34-s − 36·36-s + 6·37-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3·4-s − 1.22·6-s − 0.755·7-s − 3.53·8-s − 2·9-s − 3.91·11-s + 1.73·12-s + 1.60·14-s + 15/4·16-s + 0.242·17-s + 4.24·18-s − 0.688·19-s − 0.436·21-s + 8.31·22-s + 2.08·23-s − 2.04·24-s − 1.53·27-s − 2.26·28-s − 1.48·29-s − 0.718·31-s − 3.71·32-s − 2.26·33-s − 0.514·34-s − 6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4181066272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4181066272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | | \( 1 \) | |
| 13 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - T + 7 T^{2} - 5 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_h_af |
| 7 | $A_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 20 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_n_u |
| 11 | $A_4\times C_2$ | \( 1 + 13 T + 87 T^{2} + 357 T^{3} + 87 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.n_dj_nt |
| 17 | $A_4\times C_2$ | \( 1 - T + 35 T^{2} - 5 T^{3} + 35 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ab_bj_af |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 53 T^{2} + 115 T^{3} + 53 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.d_cb_el |
| 23 | $A_4\times C_2$ | \( 1 - 10 T + 93 T^{2} - 468 T^{3} + 93 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ak_dp_asa |
| 29 | $A_4\times C_2$ | \( 1 + 8 T + 99 T^{2} + 456 T^{3} + 99 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.i_dv_ro |
| 31 | $A_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 16 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.e_bh_q |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 39 T^{2} - 228 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_bn_aiu |
| 41 | $A_4\times C_2$ | \( 1 + 19 T + 213 T^{2} + 1671 T^{3} + 213 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.t_if_cmh |
| 43 | $A_4\times C_2$ | \( 1 + 5 T - 5 T^{2} - 409 T^{3} - 5 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_af_apt |
| 47 | $A_4\times C_2$ | \( 1 - 8 T + 97 T^{2} - 744 T^{3} + 97 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_dt_abcq |
| 53 | $A_4\times C_2$ | \( 1 - 8 T + 59 T^{2} - 280 T^{3} + 59 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ai_ch_aku |
| 59 | $A_4\times C_2$ | \( 1 + 5 T + 43 T^{2} - 249 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.f_br_ajp |
| 61 | $A_4\times C_2$ | \( 1 + 8 T + 83 T^{2} + 408 T^{3} + 83 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.i_df_ps |
| 67 | $A_4\times C_2$ | \( 1 - T + 59 T^{2} - 693 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ab_ch_abar |
| 71 | $A_4\times C_2$ | \( 1 + 12 T + 177 T^{2} + 1376 T^{3} + 177 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.m_gv_cay |
| 73 | $A_4\times C_2$ | \( 1 - 5 T + 113 T^{2} - 87 T^{3} + 113 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.af_ej_adj |
| 79 | $A_4\times C_2$ | \( 1 - 2 T + 173 T^{2} - 420 T^{3} + 173 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_gr_aqe |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 151 T^{2} + 1365 T^{3} + 151 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.h_fv_can |
| 89 | $A_4\times C_2$ | \( 1 + 21 T + 365 T^{2} + 3689 T^{3} + 365 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.v_ob_flx |
| 97 | $A_4\times C_2$ | \( 1 - 21 T + 431 T^{2} - 4361 T^{3} + 431 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.av_qp_aglt |
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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
−7.16834355207585309904813621120, −6.76383528961792590116833879953, −6.42802346281715120450619081666, −6.42071366643635983933399159760, −5.80077546308609981775068179442, −5.79217380666398102323242509885, −5.64098930841851141810680381034, −5.29989214463769026346543117783, −5.21987435590775327277108624386, −4.87111533917611815555212099377, −4.75247867183524844185312729154, −4.15479526212106839637845294357, −3.84767953248203422529685378809, −3.25328418056709824187617213026, −3.18035766043370144421042575406, −3.13748338667124480465338148832, −2.77812668550935152834710340451, −2.73171467221350810958641154211, −2.45049676273258836539100665724, −1.90670011348352381702305438522, −1.73794287773216648199898766820, −1.69423915760835648748569978530, −0.51537009904316792917569991822, −0.49707070028903754036842049686, −0.34899885718620851825160383611,
0.34899885718620851825160383611, 0.49707070028903754036842049686, 0.51537009904316792917569991822, 1.69423915760835648748569978530, 1.73794287773216648199898766820, 1.90670011348352381702305438522, 2.45049676273258836539100665724, 2.73171467221350810958641154211, 2.77812668550935152834710340451, 3.13748338667124480465338148832, 3.18035766043370144421042575406, 3.25328418056709824187617213026, 3.84767953248203422529685378809, 4.15479526212106839637845294357, 4.75247867183524844185312729154, 4.87111533917611815555212099377, 5.21987435590775327277108624386, 5.29989214463769026346543117783, 5.64098930841851141810680381034, 5.79217380666398102323242509885, 5.80077546308609981775068179442, 6.42071366643635983933399159760, 6.42802346281715120450619081666, 6.76383528961792590116833879953, 7.16834355207585309904813621120
