| L(s) = 1 | + 3·2-s − 3-s + 6·4-s + 4·5-s − 3·6-s − 3·7-s + 10·8-s − 9-s + 12·10-s − 3·11-s − 6·12-s − 9·14-s − 4·15-s + 15·16-s − 8·17-s − 3·18-s − 4·19-s + 24·20-s + 3·21-s − 9·22-s − 10·24-s + 3·25-s + 27-s − 18·28-s − 11·29-s − 12·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 3·4-s + 1.78·5-s − 1.22·6-s − 1.13·7-s + 3.53·8-s − 1/3·9-s + 3.79·10-s − 0.904·11-s − 1.73·12-s − 2.40·14-s − 1.03·15-s + 15/4·16-s − 1.94·17-s − 0.707·18-s − 0.917·19-s + 5.36·20-s + 0.654·21-s − 1.91·22-s − 2.04·24-s + 3/5·25-s + 0.192·27-s − 3.40·28-s − 2.04·29-s − 2.19·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 2 T^{2} + 2 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_c_c |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 31 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ae_n_abf |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 30 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_s_be |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 19 T^{3} + 21 p T^{4} + p^{3} T^{6} \) | 3.13.a_v_t |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 52 T^{2} + 218 T^{3} + 52 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.i_ca_ik |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 32 T^{2} + 160 T^{3} + 32 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_bg_ge |
| 29 | $S_4\times C_2$ | \( 1 + 11 T + 97 T^{2} + 530 T^{3} + 97 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.l_dt_uk |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 68 T^{2} - 256 T^{3} + 68 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ae_cq_ajw |
| 37 | $S_4\times C_2$ | \( 1 + 16 T + 176 T^{2} + 1246 T^{3} + 176 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.q_gu_bvy |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 91 T^{2} + 337 T^{3} + 91 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_dn_mz |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 120 T^{2} + 580 T^{3} + 120 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.i_eq_wi |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 78 T^{2} + 174 T^{3} + 78 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.d_da_gs |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 558 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_en_vm |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 420 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.g_en_qe |
| 61 | $S_4\times C_2$ | \( 1 + 17 T + 249 T^{2} + 2038 T^{3} + 249 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.r_jp_dak |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 89 T^{2} + 280 T^{3} + 89 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.e_dl_ku |
| 71 | $S_4\times C_2$ | \( 1 - 23 T + 352 T^{2} - 3374 T^{3} + 352 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ax_no_aezu |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 174 T^{2} + 973 T^{3} + 174 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.j_gs_bll |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 216 T^{2} + 1708 T^{3} + 216 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.u_ii_cns |
| 83 | $S_4\times C_2$ | \( 1 - 13 T + 268 T^{2} - 2134 T^{3} + 268 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.an_ki_adec |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{3} \) | 3.89.bt_bkg_qvx |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} + 1421 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ad_s_ccr |
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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
−7.07995174894440397529664125387, −6.99552205596874142890655255435, −6.69965531637372083979652321673, −6.41670565207748725546140343958, −6.19077015261040266169160141147, −6.08441627455990855397765006186, −6.03927835975592187932666947054, −5.65158272146840447568312785210, −5.38472438513691278215712649868, −5.33800093068991882679550138736, −5.01009585774606498603141833360, −4.77616767727702723406025445157, −4.48508615869088487041972114189, −4.39718791024087631701610723907, −3.84247507404148024380473000668, −3.81846518853179607319673319599, −3.27377004508679129767881557384, −3.22803660029202601142433772543, −3.03272789100574831606456072303, −2.46985673165388765052225952356, −2.32158365766062229403439902028, −2.19534051915074255172082501403, −1.78521490885736975563186356607, −1.50838745345643818061093757995, −1.37065561833906343849554782816, 0, 0, 0,
1.37065561833906343849554782816, 1.50838745345643818061093757995, 1.78521490885736975563186356607, 2.19534051915074255172082501403, 2.32158365766062229403439902028, 2.46985673165388765052225952356, 3.03272789100574831606456072303, 3.22803660029202601142433772543, 3.27377004508679129767881557384, 3.81846518853179607319673319599, 3.84247507404148024380473000668, 4.39718791024087631701610723907, 4.48508615869088487041972114189, 4.77616767727702723406025445157, 5.01009585774606498603141833360, 5.33800093068991882679550138736, 5.38472438513691278215712649868, 5.65158272146840447568312785210, 6.03927835975592187932666947054, 6.08441627455990855397765006186, 6.19077015261040266169160141147, 6.41670565207748725546140343958, 6.69965531637372083979652321673, 6.99552205596874142890655255435, 7.07995174894440397529664125387
