| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.86779753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.86779753\) |
| \(L(\frac{3}{2})\) |
|
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.ad_g_am |
| 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.ad_m_abu |
| 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.d_ci_el |
| 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.ag_cx_akl |
| 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.ad_bw_ajz |
| 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.d_en_jm |
| 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.am_fc_abrc |
| 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.y_ma_ebc |
| 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.ag_fx_abaa |
| 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.ap_eh_ayg |
| 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.p_jy_dgo |
| 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.ag_fr_abmv |
| 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 |
| 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
−7.55595351187053642979851721953, −7.28032172022254054452383148050, −6.95385519584931595159899740785, −6.66066647243147141247341440625, −6.42464228041243709799289926761, −6.26327823046298245974720299930, −6.11527136466225808308451137764, −5.61982520264766042764915625303, −5.19744907183598059437990986473, −5.18784479365775872555368119522, −4.74860295726542995863022934361, −4.62085997033475038002901758821, −4.41448494575465938035286488277, −3.91802467582255600375743306679, −3.69025954952521827178284207298, −3.50029365687952120220312176537, −3.15895914236259248250961976048, −2.99952399014876014507356054601, −2.66058598214247115968670586950, −2.16420003784692101829480577875, −2.11790545417944915247712233306, −1.72193191203450511306766954759, −1.16096781710517188131771503471, −0.990887517253212204352614077777, −0.65775892690977274096725515561,
0.65775892690977274096725515561, 0.990887517253212204352614077777, 1.16096781710517188131771503471, 1.72193191203450511306766954759, 2.11790545417944915247712233306, 2.16420003784692101829480577875, 2.66058598214247115968670586950, 2.99952399014876014507356054601, 3.15895914236259248250961976048, 3.50029365687952120220312176537, 3.69025954952521827178284207298, 3.91802467582255600375743306679, 4.41448494575465938035286488277, 4.62085997033475038002901758821, 4.74860295726542995863022934361, 5.18784479365775872555368119522, 5.19744907183598059437990986473, 5.61982520264766042764915625303, 6.11527136466225808308451137764, 6.26327823046298245974720299930, 6.42464228041243709799289926761, 6.66066647243147141247341440625, 6.95385519584931595159899740785, 7.28032172022254054452383148050, 7.55595351187053642979851721953
