| L(s) = 1 | + 3·2-s + 2·3-s + 6·4-s + 4·5-s + 6·6-s + 10·8-s − 3·9-s + 12·10-s − 2·11-s + 12·12-s + 8·13-s + 8·15-s + 15·16-s + 4·17-s − 9·18-s + 8·19-s + 24·20-s − 6·22-s + 6·23-s + 20·24-s − 25-s + 24·26-s − 10·27-s − 6·29-s + 24·30-s + 14·31-s + 21·32-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.15·3-s + 3·4-s + 1.78·5-s + 2.44·6-s + 3.53·8-s − 9-s + 3.79·10-s − 0.603·11-s + 3.46·12-s + 2.21·13-s + 2.06·15-s + 15/4·16-s + 0.970·17-s − 2.12·18-s + 1.83·19-s + 5.36·20-s − 1.27·22-s + 1.25·23-s + 4.08·24-s − 1/5·25-s + 4.70·26-s − 1.92·27-s − 1.11·29-s + 4.38·30-s + 2.51·31-s + 3.71·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(82.35438170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(82.35438170\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 7 | | \( 1 \) | |
| 71 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 10 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_h_ak |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ae_r_abm |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 21 T^{2} + 54 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_v_cc |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 51 T^{2} - 212 T^{3} + 51 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ai_bz_aie |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aem |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 75 T^{2} - 314 T^{3} + 75 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ai_cx_amc |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 17 T^{2} - 28 T^{3} + 17 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_r_abc |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 100 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_bj_dw |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 153 T^{2} - 944 T^{3} + 153 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ao_fx_abki |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 732 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.k_ed_bce |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 39 T^{2} + 84 T^{3} + 39 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ae_bn_dg |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 81 T^{2} - 424 T^{3} + 81 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ae_dd_aqi |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 149 T^{2} - 736 T^{3} + 149 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_ft_abci |
| 53 | $S_4\times C_2$ | \( 1 + 155 T^{2} + 2 T^{3} + 155 p T^{4} + p^{3} T^{6} \) | 3.53.a_fz_c |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 245 T^{2} - 2024 T^{3} + 245 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.as_jl_aczw |
| 61 | $S_4\times C_2$ | \( 1 - 12 T + 203 T^{2} - 1364 T^{3} + 203 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.am_hv_acam |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 106 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_cb_aec |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 123 T^{2} - 808 T^{3} + 123 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_et_abfc |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1496 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.am_gj_acfo |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 175 T^{2} - 258 T^{3} + 175 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ac_gt_ajy |
| 89 | $S_4\times C_2$ | \( 1 - 20 T + 339 T^{2} - 3288 T^{3} + 339 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.au_nb_aewm |
| 97 | $S_4\times C_2$ | \( 1 + 16 T + 191 T^{2} + 1404 T^{3} + 191 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.q_hj_cca |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.03557169428664867083062158397, −6.69592531163751565744841253253, −6.27037212757865658260905831378, −6.23674768642784737483659343896, −5.89277982514023504220491822261, −5.77258054136930600722171144091, −5.66009251385072619449920087212, −5.34864540129562148566533439915, −5.22250156060319635710379391063, −5.05499064171404583275446825842, −4.73932463850390108312705792863, −4.27427129809025137187720992870, −3.92862105070602496042890832308, −3.64411110372923476451323893639, −3.56428413942502740762456022442, −3.45601467983856288929126113145, −2.98129711478765337874131829105, −2.77037181072276374200138550670, −2.71268143917870491798833322322, −2.09232809584939560871292070960, −2.02078358474292940747562037442, −2.01948173205590935427498114730, −1.16828292826405261373187673957, −0.960747344520538033909680054310, −0.819508970682317472379821925269,
0.819508970682317472379821925269, 0.960747344520538033909680054310, 1.16828292826405261373187673957, 2.01948173205590935427498114730, 2.02078358474292940747562037442, 2.09232809584939560871292070960, 2.71268143917870491798833322322, 2.77037181072276374200138550670, 2.98129711478765337874131829105, 3.45601467983856288929126113145, 3.56428413942502740762456022442, 3.64411110372923476451323893639, 3.92862105070602496042890832308, 4.27427129809025137187720992870, 4.73932463850390108312705792863, 5.05499064171404583275446825842, 5.22250156060319635710379391063, 5.34864540129562148566533439915, 5.66009251385072619449920087212, 5.77258054136930600722171144091, 5.89277982514023504220491822261, 6.23674768642784737483659343896, 6.27037212757865658260905831378, 6.69592531163751565744841253253, 7.03557169428664867083062158397
