| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 3·5-s + 9·6-s − 6·7-s + 10·8-s + 6·9-s − 9·10-s − 3·11-s + 18·12-s − 9·13-s − 18·14-s − 9·15-s + 15·16-s − 12·17-s + 18·18-s − 6·19-s − 18·20-s − 18·21-s − 9·22-s − 6·23-s + 30·24-s − 6·25-s − 27·26-s + 10·27-s − 36·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s − 2.26·7-s + 3.53·8-s + 2·9-s − 2.84·10-s − 0.904·11-s + 5.19·12-s − 2.49·13-s − 4.81·14-s − 2.32·15-s + 15/4·16-s − 2.91·17-s + 4.24·18-s − 1.37·19-s − 4.02·20-s − 3.92·21-s − 1.91·22-s − 1.25·23-s + 6.12·24-s − 6/5·25-s − 5.29·26-s + 1.92·27-s − 6.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 443^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 443^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 443 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 29 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_p_bd |
| 7 | $A_4\times C_2$ | \( 1 + 6 T + 30 T^{2} + 85 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.g_be_dh |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 65 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_bh_cn |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) | 3.13.j_co_kb |
| 17 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 461 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.m_ds_rt |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 92 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.g_bh_do |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_ci_ir |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 81 T^{2} + 157 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.d_dd_gb |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 75 T^{2} - 149 T^{3} + 75 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_cx_aft |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 114 T^{2} + 443 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_ek_rb |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 511 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ds_atr |
| 43 | $A_4\times C_2$ | \( 1 + 15 T + 87 T^{2} + 353 T^{3} + 87 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.p_dj_np |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 84 T^{2} + 325 T^{3} + 84 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.j_dg_mn |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 3 p T^{2} + 873 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_gd_bhp |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 191 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ad_et_ahj |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 39 T^{2} - 563 T^{3} + 39 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ad_bn_avr |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 237 T^{2} + 1921 T^{3} + 237 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.p_jd_cvx |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 15 T^{2} + 495 T^{3} + 15 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_p_tb |
| 73 | $A_4\times C_2$ | \( 1 - 12 T + 78 T^{2} - 61 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_da_acj |
| 79 | $A_4\times C_2$ | \( 1 - 15 T + 93 T^{2} - 481 T^{3} + 93 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ap_dp_asn |
| 83 | $A_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1197 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.p_gv_bub |
| 89 | $A_4\times C_2$ | \( 1 - 15 T + 279 T^{2} - 2489 T^{3} + 279 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_kt_adrt |
| 97 | $A_4\times C_2$ | \( 1 + 15 T + 237 T^{2} + 2319 T^{3} + 237 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.p_jd_dlf |
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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
−8.287753692185403205649293212053, −7.75190794844199641894263399279, −7.68603639199466321557848990601, −7.53550462387028038484293024864, −7.04460243559184770063590286329, −6.94063113317453212074967351092, −6.60614469255252778735981543396, −6.39473312403243061185586293733, −6.33263645857543431226141097031, −6.15947140433132010462102334678, −5.34784455684696467885012920189, −5.21070321151699010917369539416, −5.02432310150829557116222910910, −4.49565312333192999943638374944, −4.39041789032172958672787369849, −4.27762933791220959941010458029, −3.72417623789224910393745600915, −3.71407675410618534092906231728, −3.48352772515654420382612919969, −2.87482491623590862795231733235, −2.87010855552443795571830189426, −2.59664212052631514486222285712, −2.10044974138004801936799174021, −1.94349738547949304704075653228, −1.85021421928200378922461459933, 0, 0, 0,
1.85021421928200378922461459933, 1.94349738547949304704075653228, 2.10044974138004801936799174021, 2.59664212052631514486222285712, 2.87010855552443795571830189426, 2.87482491623590862795231733235, 3.48352772515654420382612919969, 3.71407675410618534092906231728, 3.72417623789224910393745600915, 4.27762933791220959941010458029, 4.39041789032172958672787369849, 4.49565312333192999943638374944, 5.02432310150829557116222910910, 5.21070321151699010917369539416, 5.34784455684696467885012920189, 6.15947140433132010462102334678, 6.33263645857543431226141097031, 6.39473312403243061185586293733, 6.60614469255252778735981543396, 6.94063113317453212074967351092, 7.04460243559184770063590286329, 7.53550462387028038484293024864, 7.68603639199466321557848990601, 7.75190794844199641894263399279, 8.287753692185403205649293212053
