| L(s) = 1 | − 2·2-s − 3-s − 4-s − 7·5-s + 2·6-s − 3·7-s + 5·8-s − 6·9-s + 14·10-s − 5·11-s + 12-s − 13-s + 6·14-s + 7·15-s − 16-s − 8·17-s + 12·18-s − 3·19-s + 7·20-s + 3·21-s + 10·22-s − 5·24-s + 20·25-s + 2·26-s + 8·27-s + 3·28-s − 29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 1/2·4-s − 3.13·5-s + 0.816·6-s − 1.13·7-s + 1.76·8-s − 2·9-s + 4.42·10-s − 1.50·11-s + 0.288·12-s − 0.277·13-s + 1.60·14-s + 1.80·15-s − 1/4·16-s − 1.94·17-s + 2.82·18-s − 0.688·19-s + 1.56·20-s + 0.654·21-s + 2.13·22-s − 1.02·24-s + 4·25-s + 0.392·26-s + 1.53·27-s + 0.566·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3442951 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3442951 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 151 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_f_h |
| 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.b_h_f |
| 5 | $A_4\times C_2$ | \( 1 + 7 T + 29 T^{2} + 77 T^{3} + 29 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.h_bd_cz |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) | 3.7.d_y_br |
| 11 | $A_4\times C_2$ | \( 1 + 5 T + 32 T^{2} + 97 T^{3} + 32 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.f_bg_dt |
| 13 | $A_4\times C_2$ | \( 1 + T + 23 T^{2} + 3 p T^{3} + 23 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.13.b_x_bn |
| 17 | $A_4\times C_2$ | \( 1 + 8 T + 56 T^{2} + 229 T^{3} + 56 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.i_ce_iv |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 11 T^{2} - 25 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.d_l_az |
| 23 | $A_4\times C_2$ | \( 1 + 48 T^{2} - 7 T^{3} + 48 p T^{4} + p^{3} T^{6} \) | 3.23.a_bw_ah |
| 29 | $C_6$ | \( 1 + T + 15 T^{2} + 99 T^{3} + 15 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.29.b_p_dv |
| 31 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} + 19 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.31.b_cl_t |
| 37 | $A_4\times C_2$ | \( 1 - 13 T + 151 T^{2} - 991 T^{3} + 151 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.an_fv_abmd |
| 41 | $A_4\times C_2$ | \( 1 + 21 T + 242 T^{2} + 1813 T^{3} + 242 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.v_ji_crt |
| 43 | $A_4\times C_2$ | \( 1 - 16 T + 170 T^{2} - 1179 T^{3} + 170 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aq_go_abtj |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 32 T^{2} + 277 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ad_bg_kr |
| 53 | $A_4\times C_2$ | \( 1 + 8 T + 136 T^{2} + 651 T^{3} + 136 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.i_fg_zb |
| 59 | $A_4\times C_2$ | \( 1 + T + 77 T^{2} + 299 T^{3} + 77 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.59.b_cz_ln |
| 61 | $A_4\times C_2$ | \( 1 + T + 125 T^{2} + 135 T^{3} + 125 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.61.b_ev_ff |
| 67 | $A_4\times C_2$ | \( 1 + T + 31 T^{2} + 175 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.67.b_bf_gt |
| 71 | $A_4\times C_2$ | \( 1 + 14 T + 164 T^{2} + 1099 T^{3} + 164 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.o_gi_bqh |
| 73 | $A_4\times C_2$ | \( 1 + T + 154 T^{2} - 23 T^{3} + 154 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.73.b_fy_ax |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 149 T^{2} + 181 T^{3} + 149 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_ft_gz |
| 83 | $A_4\times C_2$ | \( 1 + 3 T + 224 T^{2} + 5 p T^{3} + 224 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.d_iq_pz |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) | 3.89.bk_bax_may |
| 97 | $A_4\times C_2$ | \( 1 + 228 T^{2} + 189 T^{3} + 228 p T^{4} + p^{3} T^{6} \) | 3.97.a_iu_hh |
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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
−12.17216837269419025187999821068, −11.56611308078878977421157428617, −11.36837301431931730265403324258, −11.14734889176512569971637913916, −11.08917759540738881127777253995, −10.45629811529634858964459711111, −10.15144443100940502362675145278, −9.635425937149745719195173202863, −9.321035124353913961411103214602, −8.963339421493218907695425541432, −8.443379859838028181513847224563, −8.412269380379253427330595278141, −8.181564201699652198607370507874, −7.73456457059618442474646554773, −7.58232614960246939882581076545, −6.81751110235975571105250679139, −6.64119553276885441127218537456, −5.80335896985344554077176299323, −5.71024005782389566270827522333, −4.77952016363445828297841514949, −4.56006782750996330048017116480, −4.21740718778131784248779329532, −3.59639775933490090524444896077, −3.01121597819291387043173268667, −2.67245042091945515097468418581, 0, 0, 0,
2.67245042091945515097468418581, 3.01121597819291387043173268667, 3.59639775933490090524444896077, 4.21740718778131784248779329532, 4.56006782750996330048017116480, 4.77952016363445828297841514949, 5.71024005782389566270827522333, 5.80335896985344554077176299323, 6.64119553276885441127218537456, 6.81751110235975571105250679139, 7.58232614960246939882581076545, 7.73456457059618442474646554773, 8.181564201699652198607370507874, 8.412269380379253427330595278141, 8.443379859838028181513847224563, 8.963339421493218907695425541432, 9.321035124353913961411103214602, 9.635425937149745719195173202863, 10.15144443100940502362675145278, 10.45629811529634858964459711111, 11.08917759540738881127777253995, 11.14734889176512569971637913916, 11.36837301431931730265403324258, 11.56611308078878977421157428617, 12.17216837269419025187999821068
