| L(s) = 1 | − 3-s − 2·9-s + 6·11-s + 7·13-s − 8·17-s − 8·19-s + 3·23-s − 5·25-s − 3·27-s + 5·29-s − 5·31-s − 6·33-s + 8·37-s − 7·39-s + 11·41-s − 4·43-s − 5·47-s + 8·51-s + 4·53-s + 8·57-s − 4·59-s − 14·61-s + 28·67-s − 3·69-s + 15·71-s + 73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 1.80·11-s + 1.94·13-s − 1.94·17-s − 1.83·19-s + 0.625·23-s − 25-s − 0.577·27-s + 0.928·29-s − 0.898·31-s − 1.04·33-s + 1.31·37-s − 1.12·39-s + 1.71·41-s − 0.609·43-s − 0.729·47-s + 1.12·51-s + 0.549·53-s + 1.05·57-s − 0.520·59-s − 1.79·61-s + 3.42·67-s − 0.361·69-s + 1.78·71-s + 0.117·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.809921935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.809921935\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_d_i |
| 5 | $D_{6}$ | \( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) | 3.5.a_f_i |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.11.ag_bt_afk |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 47 T^{2} - 174 T^{3} + 47 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ah_bv_ags |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 208 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.i_cb_ia |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 45 T^{2} + 224 T^{3} + 45 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_bt_iq |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 282 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.af_dj_akw |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 95 T^{2} + 300 T^{3} + 95 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.f_dr_lo |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 107 T^{2} - 576 T^{3} + 107 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_ed_awe |
| 41 | $S_4\times C_2$ | \( 1 - 11 T + 139 T^{2} - 834 T^{3} + 139 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.al_fj_abgc |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 472 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_bh_se |
| 47 | $S_4\times C_2$ | \( 1 + 5 T + 87 T^{2} + 348 T^{3} + 87 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.f_dj_nk |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} + 216 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ae_br_ii |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 115 T^{2} + 240 T^{3} + 115 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.e_el_jg |
| 61 | $S_4\times C_2$ | \( 1 + 14 T + 229 T^{2} + 1688 T^{3} + 229 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.o_iv_cmy |
| 67 | $S_4\times C_2$ | \( 1 - 28 T + 429 T^{2} - 4264 T^{3} + 429 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.abc_qn_agia |
| 71 | $S_4\times C_2$ | \( 1 - 15 T + 197 T^{2} - 1466 T^{3} + 197 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ap_hp_acek |
| 73 | $S_4\times C_2$ | \( 1 - T + 99 T^{2} - 46 T^{3} + 99 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ab_dv_abu |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 121 T^{2} + 8 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_er_i |
| 83 | $S_4\times C_2$ | \( 1 + 20 T + 357 T^{2} + 3480 T^{3} + 357 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.u_nt_fdw |
| 89 | $S_4\times C_2$ | \( 1 - 12 T + 141 T^{2} - 656 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.am_fl_azg |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 309 T^{2} - 2952 T^{3} + 309 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.as_lx_aejo |
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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
−6.58903444676899890735024609593, −6.42164026368765786086678163903, −6.34849792503849043212726661694, −6.31479912835078099459501219002, −5.90615527568160536284651960500, −5.86813022159678052915125394474, −5.70502150716408377920587576426, −5.01589330544242038120450728539, −5.00161605333900895007839798961, −4.66631623336021252508917734058, −4.55278896907989136657153467415, −4.12236297294782682372179819719, −3.97121432898409082143731665425, −3.71028747023378903555863820815, −3.60850465568836321302489352564, −3.44108497459857992892115864890, −2.79656462918219664631820952278, −2.61685275242862959240871786472, −2.29905987265840591313600120726, −1.92938019726243803360249835174, −1.74148609665069346545315634205, −1.56731166795178467529417559464, −0.868950508290354562800942542319, −0.64013437507452349748466861874, −0.42498408055327924562984213522,
0.42498408055327924562984213522, 0.64013437507452349748466861874, 0.868950508290354562800942542319, 1.56731166795178467529417559464, 1.74148609665069346545315634205, 1.92938019726243803360249835174, 2.29905987265840591313600120726, 2.61685275242862959240871786472, 2.79656462918219664631820952278, 3.44108497459857992892115864890, 3.60850465568836321302489352564, 3.71028747023378903555863820815, 3.97121432898409082143731665425, 4.12236297294782682372179819719, 4.55278896907989136657153467415, 4.66631623336021252508917734058, 5.00161605333900895007839798961, 5.01589330544242038120450728539, 5.70502150716408377920587576426, 5.86813022159678052915125394474, 5.90615527568160536284651960500, 6.31479912835078099459501219002, 6.34849792503849043212726661694, 6.42164026368765786086678163903, 6.58903444676899890735024609593
