| L(s) = 1 | − 3·4-s − 6·5-s − 8-s − 3·11-s + 3·13-s + 3·16-s + 3·17-s + 9·19-s + 18·20-s + 12·25-s + 3·29-s + 9·31-s + 6·32-s + 6·40-s − 9·41-s + 9·44-s + 3·47-s − 9·52-s − 9·53-s + 18·55-s + 12·61-s + 2·64-s − 18·65-s − 9·68-s + 9·71-s + 6·73-s − 27·76-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2.68·5-s − 0.353·8-s − 0.904·11-s + 0.832·13-s + 3/4·16-s + 0.727·17-s + 2.06·19-s + 4.02·20-s + 12/5·25-s + 0.557·29-s + 1.61·31-s + 1.06·32-s + 0.948·40-s − 1.40·41-s + 1.35·44-s + 0.437·47-s − 1.24·52-s − 1.23·53-s + 2.42·55-s + 1.53·61-s + 1/4·64-s − 2.23·65-s − 1.09·68-s + 1.06·71-s + 0.702·73-s − 3.09·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533958260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533958260\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T^{2} + T^{3} + 3 p T^{4} + p^{3} T^{6} \) | 3.2.a_d_b |
| 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 61 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.g_y_cj |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 79 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_bh_adb |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} + 25 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_p_z |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 75 T^{2} - 351 T^{3} + 75 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.aj_cx_ann |
| 23 | $A_4\times C_2$ | \( 1 + 12 T^{2} + 107 T^{3} + 12 p T^{4} + p^{3} T^{6} \) | 3.23.a_m_ed |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 225 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_bz_air |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 577 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.aj_en_awf |
| 37 | $A_4\times C_2$ | \( 1 + 75 T^{2} + 72 T^{3} + 75 p T^{4} + p^{3} T^{6} \) | 3.37.a_cx_cu |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 129 T^{2} + 739 T^{3} + 129 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.j_ez_bcl |
| 43 | $A_4\times C_2$ | \( 1 + 120 T^{2} + 9 T^{3} + 120 p T^{4} + p^{3} T^{6} \) | 3.43.a_eq_j |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 63 T^{2} + 41 T^{3} + 63 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ad_cl_bp |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 105 T^{2} + 495 T^{3} + 105 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_eb_tb |
| 59 | $A_4\times C_2$ | \( 1 + 84 T^{2} - 19 T^{3} + 84 p T^{4} + p^{3} T^{6} \) | 3.59.a_dg_at |
| 61 | $A_4\times C_2$ | \( 1 - 12 T + 147 T^{2} - 1488 T^{3} + 147 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.am_fr_acfg |
| 67 | $A_4\times C_2$ | \( 1 + 189 T^{2} - 8 T^{3} + 189 p T^{4} + p^{3} T^{6} \) | 3.67.a_hh_ai |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 123 T^{2} - 477 T^{3} + 123 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_et_asj |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 228 T^{2} - 877 T^{3} + 228 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_iu_abht |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 511 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_et_tr |
| 83 | $A_4\times C_2$ | \( 1 - 15 T + 267 T^{2} - 2223 T^{3} + 267 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ap_kh_adhn |
| 89 | $A_4\times C_2$ | \( 1 - 15 T + 339 T^{2} - 2781 T^{3} + 339 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_nb_aecz |
| 97 | $A_4\times C_2$ | \( 1 - 45 T + 963 T^{2} - 12059 T^{3} + 963 p T^{4} - 45 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.abt_blb_arvv |
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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
−7.58456404248626323745176190210, −7.36511670665740333500721079864, −6.78017054491671848832871235427, −6.66925663501354378512531442544, −6.39808590860196296766314585295, −6.01591855783966894567420325996, −5.93232522706278188429980235976, −5.33884696558268718318998715793, −5.20728901094009184833310375840, −5.09047683150218358335323993348, −4.74353337813995774623406567979, −4.48920064566228660878637927372, −4.44218521778173120821058517570, −3.79878446224056054598964997368, −3.77291136948482726006282236313, −3.64411457229626365721784870067, −3.25807692096509793276710904700, −3.06638684022604061968399874352, −2.90638140575029495938289971301, −2.25077359487707979923487583511, −1.94824979982848695108338011261, −1.35588855902467960567044463816, −0.63830559128734014288146320648, −0.61964686365068318668732901253, −0.56955130176234257065871346941,
0.56955130176234257065871346941, 0.61964686365068318668732901253, 0.63830559128734014288146320648, 1.35588855902467960567044463816, 1.94824979982848695108338011261, 2.25077359487707979923487583511, 2.90638140575029495938289971301, 3.06638684022604061968399874352, 3.25807692096509793276710904700, 3.64411457229626365721784870067, 3.77291136948482726006282236313, 3.79878446224056054598964997368, 4.44218521778173120821058517570, 4.48920064566228660878637927372, 4.74353337813995774623406567979, 5.09047683150218358335323993348, 5.20728901094009184833310375840, 5.33884696558268718318998715793, 5.93232522706278188429980235976, 6.01591855783966894567420325996, 6.39808590860196296766314585295, 6.66925663501354378512531442544, 6.78017054491671848832871235427, 7.36511670665740333500721079864, 7.58456404248626323745176190210
