| L(s) = 1 | − 3-s − 3·5-s + 3·7-s − 3·9-s − 6·11-s − 13-s + 3·15-s + 5·17-s − 2·19-s − 3·21-s − 11·23-s + 6·25-s − 27-s − 3·29-s − 31-s + 6·33-s − 9·35-s + 14·37-s + 39-s − 19·43-s + 9·45-s − 10·47-s − 11·49-s − 5·51-s + 9·53-s + 18·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.13·7-s − 9-s − 1.80·11-s − 0.277·13-s + 0.774·15-s + 1.21·17-s − 0.458·19-s − 0.654·21-s − 2.29·23-s + 6/5·25-s − 0.192·27-s − 0.557·29-s − 0.179·31-s + 1.04·33-s − 1.52·35-s + 2.30·37-s + 0.160·39-s − 2.89·43-s + 1.34·45-s − 1.45·47-s − 1.57·49-s − 0.700·51-s + 1.23·53-s + 2.42·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + 8 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_e_i |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 20 T^{2} - 38 T^{3} + 20 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_u_abm |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 29 T^{2} + 100 T^{3} + 29 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bd_dw |
| 13 | $S_4\times C_2$ | \( 1 + T + 34 T^{2} + 28 T^{3} + 34 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.13.b_bi_bc |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 28 T^{2} - 114 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.af_bc_aek |
| 19 | $S_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 148 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_ad_afs |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 94 T^{2} + 522 T^{3} + 94 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.l_dq_uc |
| 31 | $S_4\times C_2$ | \( 1 + T + 74 T^{2} + 30 T^{3} + 74 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.31.b_cw_be |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 155 T^{2} - 1068 T^{3} + 155 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ao_fz_abpc |
| 41 | $S_4\times C_2$ | \( 1 + 107 T^{2} + 8 T^{3} + 107 p T^{4} + p^{3} T^{6} \) | 3.41.a_ed_i |
| 43 | $S_4\times C_2$ | \( 1 + 19 T + 230 T^{2} + 1740 T^{3} + 230 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.t_iw_coy |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 149 T^{2} + 908 T^{3} + 149 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.k_ft_biy |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 150 T^{2} - 900 T^{3} + 150 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_fu_abiq |
| 59 | $S_4\times C_2$ | \( 1 + T + 40 T^{2} - 6 T^{3} + 40 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.59.b_bo_ag |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 68 T^{2} - 420 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ah_cq_aqe |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 121 T^{2} + 944 T^{3} + 121 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_er_bki |
| 71 | $S_4\times C_2$ | \( 1 + 16 T + 197 T^{2} + 1760 T^{3} + 197 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.q_hp_cps |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 68 T^{2} + 82 T^{3} + 68 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_cq_de |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 196 T^{2} + 898 T^{3} + 196 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.h_ho_bio |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 261 T^{2} + 1596 T^{3} + 261 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.k_kb_cjk |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 379 T^{2} + 3744 T^{3} + 379 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_op_foa |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 236 T^{2} + 1634 T^{3} + 236 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.j_jc_ckw |
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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
−8.315096360812951787817330946009, −8.216016747727949413247994822561, −7.79185177576285762709687639765, −7.72910147355489703037701989751, −7.61446426977454530957297556422, −7.03189182917678455284499066892, −6.97220408246874266666177501907, −6.50552074485786732354253007220, −6.09002632698862816685737945971, −6.08251934172827433262852935396, −5.48752470563740399073650462925, −5.46343285579612545392812532937, −5.32873417586353388178986597043, −4.80120928138482363700496745374, −4.78127031523252825495546177822, −4.26090051952217487449287753366, −4.05807862402089756947558952971, −3.71313795257993192712601048438, −3.51457714571073270727550499443, −2.79939233125817263254683959022, −2.76602978150688915331095111334, −2.63006684187719674399850534279, −1.78671930937933840668810394386, −1.65885597051326279955354370982, −1.18026854670003186299914407663, 0, 0, 0,
1.18026854670003186299914407663, 1.65885597051326279955354370982, 1.78671930937933840668810394386, 2.63006684187719674399850534279, 2.76602978150688915331095111334, 2.79939233125817263254683959022, 3.51457714571073270727550499443, 3.71313795257993192712601048438, 4.05807862402089756947558952971, 4.26090051952217487449287753366, 4.78127031523252825495546177822, 4.80120928138482363700496745374, 5.32873417586353388178986597043, 5.46343285579612545392812532937, 5.48752470563740399073650462925, 6.08251934172827433262852935396, 6.09002632698862816685737945971, 6.50552074485786732354253007220, 6.97220408246874266666177501907, 7.03189182917678455284499066892, 7.61446426977454530957297556422, 7.72910147355489703037701989751, 7.79185177576285762709687639765, 8.216016747727949413247994822561, 8.315096360812951787817330946009
