| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s − 4·8-s − 2·9-s + 2·10-s + 3·12-s + 8·13-s − 15-s + 3·16-s + 8·17-s + 4·18-s − 3·20-s + 7·23-s − 4·24-s − 6·25-s − 16·26-s + 3·27-s + 2·30-s + 13·31-s − 6·32-s − 16·34-s − 6·36-s − 17·37-s + 8·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s − 1.41·8-s − 2/3·9-s + 0.632·10-s + 0.866·12-s + 2.21·13-s − 0.258·15-s + 3/4·16-s + 1.94·17-s + 0.942·18-s − 0.670·20-s + 1.45·23-s − 0.816·24-s − 6/5·25-s − 3.13·26-s + 0.577·27-s + 0.365·30-s + 2.33·31-s − 1.06·32-s − 2.74·34-s − 36-s − 2.79·37-s + 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.801925569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.801925569\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | | \( 1 \) | |
| 11 | | \( 1 \) | |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_b_a |
| 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.ab_d_ai |
| 5 | $S_4\times C_2$ | \( 1 + T + 7 T^{2} + 6 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.5.b_h_g |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ai_bp_afo |
| 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.ai_cb_aia |
| 19 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 64 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.19.a_r_acm |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 77 T^{2} - 314 T^{3} + 77 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ah_cz_amc |
| 29 | $S_4\times C_2$ | \( 1 + 47 T^{2} - 64 T^{3} + 47 p T^{4} + p^{3} T^{6} \) | 3.29.a_bv_acm |
| 31 | $S_4\times C_2$ | \( 1 - 13 T + 143 T^{2} - 864 T^{3} + 143 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.an_fn_abhg |
| 37 | $S_4\times C_2$ | \( 1 + 17 T + 183 T^{2} + 1274 T^{3} + 183 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.r_hb_bxa |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 189 T^{2} - 1392 T^{3} + 189 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aq_hh_acbo |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 101 T^{2} + 312 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_dx_ma |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 127 T^{2} + 384 T^{3} + 127 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.e_ex_ou |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 3 p T^{2} + 996 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.k_gd_bmi |
| 59 | $S_4\times C_2$ | \( 1 + T + 171 T^{2} + 120 T^{3} + 171 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.59.b_gp_eq |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 249 T^{2} + 2032 T^{3} + 249 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.q_jp_dae |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 113 T^{2} - 22 T^{3} + 113 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.d_ej_aw |
| 71 | $S_4\times C_2$ | \( 1 - 5 T + 5 T^{2} + 770 T^{3} + 5 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.af_f_bdq |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 285 T^{2} - 2416 T^{3} + 285 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aq_kz_adoy |
| 79 | $S_4\times C_2$ | \( 1 + 28 T + 465 T^{2} + 4936 T^{3} + 465 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.bc_rx_hhw |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 193 T^{2} - 1392 T^{3} + 193 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ai_hl_acbo |
| 89 | $S_4\times C_2$ | \( 1 - 21 T + 371 T^{2} - 3838 T^{3} + 371 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.av_oh_afrq |
| 97 | $S_4\times C_2$ | \( 1 - 11 T + 259 T^{2} - 1682 T^{3} + 259 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.al_jz_acms |
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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
−7.42843258373723363994676970846, −7.03694123958476851385411642419, −6.60866703987631687593349306061, −6.53686751554898612050055053115, −6.28981654156943420911134746088, −5.97418502396502564632088931388, −5.94088068869327464025464372508, −5.63215216965030861164973721922, −5.19797355413528827064907491362, −4.97935735739433993547917264153, −4.80595727983828368423076209198, −4.44931249462997610935806203639, −3.97099707411650684804518533452, −3.81071019833881417455299467036, −3.47321808347612200726010995262, −3.15038888554770456422657705210, −3.07276612540714547155374783346, −3.03603504392090148691326895857, −2.32492212786160323946511534204, −2.24528511081675814040599468592, −1.62347841224271599684419824872, −1.37183138521920715636493417454, −1.23799549330227407322031050418, −0.65504734005695103753546160553, −0.45381627401680014378612570279,
0.45381627401680014378612570279, 0.65504734005695103753546160553, 1.23799549330227407322031050418, 1.37183138521920715636493417454, 1.62347841224271599684419824872, 2.24528511081675814040599468592, 2.32492212786160323946511534204, 3.03603504392090148691326895857, 3.07276612540714547155374783346, 3.15038888554770456422657705210, 3.47321808347612200726010995262, 3.81071019833881417455299467036, 3.97099707411650684804518533452, 4.44931249462997610935806203639, 4.80595727983828368423076209198, 4.97935735739433993547917264153, 5.19797355413528827064907491362, 5.63215216965030861164973721922, 5.94088068869327464025464372508, 5.97418502396502564632088931388, 6.28981654156943420911134746088, 6.53686751554898612050055053115, 6.60866703987631687593349306061, 7.03694123958476851385411642419, 7.42843258373723363994676970846
