| L(s) = 1 | − 3·3-s − 3·7-s + 3·9-s − 3·11-s + 3·13-s + 3·17-s + 3·19-s + 9·21-s − 6·23-s − 3·27-s + 12·29-s − 3·31-s + 9·33-s − 12·37-s − 9·39-s + 6·41-s + 3·43-s − 12·47-s − 3·49-s − 9·51-s + 9·53-s − 9·57-s + 24·59-s − 3·61-s − 9·63-s − 6·67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 9-s − 0.904·11-s + 0.832·13-s + 0.727·17-s + 0.688·19-s + 1.96·21-s − 1.25·23-s − 0.577·27-s + 2.22·29-s − 0.538·31-s + 1.56·33-s − 1.97·37-s − 1.44·39-s + 0.937·41-s + 0.457·43-s − 1.75·47-s − 3/7·49-s − 1.26·51-s + 1.23·53-s − 1.19·57-s + 3.12·59-s − 0.384·61-s − 1.13·63-s − 0.733·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041222732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041222732\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 2 p T^{2} + 4 p T^{3} + 2 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_g_m |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 12 T^{2} + 46 T^{3} + 12 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_m_bu |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} - 63 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_s_acl |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 42 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_g_bq |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) | 3.19.ad_ci_ael |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 271 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cx_kl |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 123 T^{2} - 727 T^{3} + 123 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.am_et_abbz |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 48 T^{2} + 259 T^{3} + 48 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.d_bw_jz |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 138 T^{2} + 884 T^{3} + 138 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.m_fi_bia |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 120 T^{2} - 472 T^{3} + 120 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_eq_ase |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 117 T^{2} - 246 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ad_en_ajm |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 1120 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.m_fc_brc |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 90 T^{2} - 412 T^{3} + 90 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_dm_apw |
| 59 | $S_4\times C_2$ | \( 1 - 24 T + 312 T^{2} - 2732 T^{3} + 312 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ay_ma_aebc |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 72 T^{2} + 20 T^{3} + 72 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.d_cu_u |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 153 T^{2} + 676 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_fx_baa |
| 71 | $S_4\times C_2$ | \( 1 + 15 T + 111 T^{2} + 630 T^{3} + 111 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.p_eh_yg |
| 73 | $S_4\times C_2$ | \( 1 - 33 T + 570 T^{2} - 6002 T^{3} + 570 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.abh_vy_aiww |
| 79 | $S_4\times C_2$ | \( 1 - 15 T + 258 T^{2} - 2198 T^{3} + 258 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ap_jy_adgo |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 147 T^{2} + 1009 T^{3} + 147 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.g_fr_bmv |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 165 T^{2} - 615 T^{3} + 165 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_gj_axr |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 171 T^{2} - 1015 T^{3} + 171 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.as_gp_abnb |
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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
−7.83163558738275652358828872145, −7.78878792613622072607170191082, −7.54935573010258722821344517853, −7.18846712844719106028004406160, −6.80141232095902593283783879395, −6.60965751098201312542657254621, −6.44515523634209317910223154526, −6.02537138343692102910632796769, −5.93446185690217601149998068165, −5.83876809321765608876647303330, −5.25885239032723166609115740682, −5.13557283856497369886046392266, −5.01496271545194156174451191348, −4.69260456036960458871851970380, −4.14346926658905974631562136302, −3.89767738170431294718644026189, −3.41057624800367008856391038257, −3.30154524532247742310462105086, −3.21353832919645187963527665035, −2.35727785110887620703344918412, −2.26480371411211567590358159617, −1.81984936627573436762887754093, −1.11422364775826936943401366603, −0.63506251583479429433550521403, −0.44674979713514283605877692824,
0.44674979713514283605877692824, 0.63506251583479429433550521403, 1.11422364775826936943401366603, 1.81984936627573436762887754093, 2.26480371411211567590358159617, 2.35727785110887620703344918412, 3.21353832919645187963527665035, 3.30154524532247742310462105086, 3.41057624800367008856391038257, 3.89767738170431294718644026189, 4.14346926658905974631562136302, 4.69260456036960458871851970380, 5.01496271545194156174451191348, 5.13557283856497369886046392266, 5.25885239032723166609115740682, 5.83876809321765608876647303330, 5.93446185690217601149998068165, 6.02537138343692102910632796769, 6.44515523634209317910223154526, 6.60965751098201312542657254621, 6.80141232095902593283783879395, 7.18846712844719106028004406160, 7.54935573010258722821344517853, 7.78878792613622072607170191082, 7.83163558738275652358828872145
