| L(s) = 1 | − 3·3-s + 6·9-s − 5·11-s − 2·13-s + 5·17-s − 5·19-s + 23-s − 10·27-s + 3·29-s + 3·31-s + 15·33-s + 6·39-s − 18·41-s + 16·43-s + 2·47-s − 13·49-s − 15·51-s − 3·53-s + 15·57-s + 24·59-s − 12·61-s + 29·67-s − 3·69-s − 6·71-s + 8·73-s − 8·79-s + 15·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 1.50·11-s − 0.554·13-s + 1.21·17-s − 1.14·19-s + 0.208·23-s − 1.92·27-s + 0.557·29-s + 0.538·31-s + 2.61·33-s + 0.960·39-s − 2.81·41-s + 2.43·43-s + 0.291·47-s − 1.85·49-s − 2.10·51-s − 0.412·53-s + 1.98·57-s + 3.12·59-s − 1.53·61-s + 3.54·67-s − 0.361·69-s − 0.712·71-s + 0.936·73-s − 0.900·79-s + 5/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 13 T^{2} - 4 T^{3} + 13 p T^{4} + p^{3} T^{6} \) | 3.7.a_n_ae |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 36 T^{2} + 103 T^{3} + 36 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.f_bk_dz |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 21 T^{2} + 70 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.c_v_cs |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 40 T^{2} - 173 T^{3} + 40 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.af_bo_agr |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 44 T^{2} + 181 T^{3} + 44 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.f_bs_gz |
| 23 | $S_4\times C_2$ | \( 1 - T + 20 T^{2} + 101 T^{3} + 20 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ab_u_dx |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 24 T^{2} + 69 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_y_cr |
| 37 | $S_4\times C_2$ | \( 1 + 73 T^{2} + 86 T^{3} + 73 p T^{4} + p^{3} T^{6} \) | 3.37.a_cv_di |
| 41 | $S_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 662 T^{3} + 127 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.s_ex_zm |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 193 T^{2} - 1392 T^{3} + 193 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aq_hl_acbo |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 196 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ac_er_aho |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 124 T^{2} + 195 T^{3} + 124 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.d_eu_hn |
| 59 | $S_4\times C_2$ | \( 1 - 24 T + 325 T^{2} - 2998 T^{3} + 325 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ay_mn_aeli |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 199 T^{2} + 1368 T^{3} + 199 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_hr_caq |
| 67 | $S_4\times C_2$ | \( 1 - 29 T + 412 T^{2} - 3897 T^{3} + 412 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.abd_pw_aftx |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 193 T^{2} + 764 T^{3} + 193 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.g_hl_bdk |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 211 T^{2} - 1056 T^{3} + 211 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ai_id_aboq |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 31 T^{2} - 510 T^{3} + 31 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.i_bf_atq |
| 83 | $S_4\times C_2$ | \( 1 + 19 T + 344 T^{2} + 3213 T^{3} + 344 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.t_ng_etp |
| 89 | $S_4\times C_2$ | \( 1 + 17 T + 314 T^{2} + 3059 T^{3} + 314 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.r_mc_enr |
| 97 | $S_4\times C_2$ | \( 1 + 17 T + 226 T^{2} + 2181 T^{3} + 226 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.r_is_dfx |
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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
−7.05253254049223060308533708881, −6.79822957351552768413705694422, −6.68888929489796509594068427207, −6.64589951824960405501542024892, −6.13705925059015201917987217145, −5.95331007464350321655430170552, −5.70371458248687966894143766642, −5.47206633465311711979474823699, −5.30779501046563223477825656852, −5.29039373978434461983581070805, −4.81806876255509662594645141165, −4.62391282877875530425430365780, −4.62070725668162641564162962937, −4.00673111597917875836809944326, −3.92329515302559605972157531244, −3.84041751767292999672541330338, −3.10429410426105606652106967065, −3.06644889897971943243269149675, −2.90851405399184208306949236781, −2.24823154317755692587466973032, −2.15856600204699639950180475872, −2.08951152167661879438093231323, −1.18856302618762972219282118363, −1.15926741657025899105808825143, −1.02938230697506083335543850570, 0, 0, 0,
1.02938230697506083335543850570, 1.15926741657025899105808825143, 1.18856302618762972219282118363, 2.08951152167661879438093231323, 2.15856600204699639950180475872, 2.24823154317755692587466973032, 2.90851405399184208306949236781, 3.06644889897971943243269149675, 3.10429410426105606652106967065, 3.84041751767292999672541330338, 3.92329515302559605972157531244, 4.00673111597917875836809944326, 4.62070725668162641564162962937, 4.62391282877875530425430365780, 4.81806876255509662594645141165, 5.29039373978434461983581070805, 5.30779501046563223477825656852, 5.47206633465311711979474823699, 5.70371458248687966894143766642, 5.95331007464350321655430170552, 6.13705925059015201917987217145, 6.64589951824960405501542024892, 6.68888929489796509594068427207, 6.79822957351552768413705694422, 7.05253254049223060308533708881
