| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 3·5-s − 9·6-s − 3·7-s + 10·8-s + 6·9-s − 9·10-s − 18·12-s − 9·13-s − 9·14-s + 9·15-s + 15·16-s − 6·17-s + 18·18-s − 3·19-s − 18·20-s + 9·21-s − 6·23-s − 30·24-s − 6·25-s − 27·26-s − 10·27-s − 18·28-s − 3·29-s + 27·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s − 2.84·10-s − 5.19·12-s − 2.49·13-s − 2.40·14-s + 2.32·15-s + 15/4·16-s − 1.45·17-s + 4.24·18-s − 0.688·19-s − 4.02·20-s + 1.96·21-s − 1.25·23-s − 6.12·24-s − 6/5·25-s − 5.29·26-s − 1.92·27-s − 3.40·28-s − 0.557·29-s + 4.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 223^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 223^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 223 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 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 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 41 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_v_bp |
| 11 | $A_4\times C_2$ | \( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \) | 3.11.a_y_aj |
| 13 | $A_4\times C_2$ | \( 1 + 9 T + 57 T^{2} + 243 T^{3} + 57 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.j_cf_jj |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 131 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.g_bk_fb |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 111 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.d_cf_eh |
| 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 | $A_4\times C_2$ | \( 1 + 3 T + 27 T^{2} - 59 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.d_bb_ach |
| 31 | $A_4\times C_2$ | \( 1 + 9 T + 81 T^{2} + 379 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.j_dd_op |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 12 T^{2} - 203 T^{3} + 12 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_m_ahv |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 96 T^{2} + 333 T^{3} + 96 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_ds_mv |
| 43 | $A_4\times C_2$ | \( 1 + 102 T^{2} + 27 T^{3} + 102 p T^{4} + p^{3} T^{6} \) | 3.43.a_dy_bb |
| 47 | $A_4\times C_2$ | \( 1 + 15 T + 123 T^{2} + 781 T^{3} + 123 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.p_et_beb |
| 53 | $A_4\times C_2$ | \( 1 + 9 T + 57 T^{2} + 145 T^{3} + 57 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_cf_fp |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 711 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_gm_abbj |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 237 T^{2} + 1833 T^{3} + 237 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.p_jd_csn |
| 67 | $A_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 653 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.d_cf_zd |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) | 3.71.aj_jg_abyf |
| 73 | $A_4\times C_2$ | \( 1 + 12 T + 246 T^{2} + 1769 T^{3} + 246 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.m_jm_cqb |
| 79 | $A_4\times C_2$ | \( 1 + 216 T^{2} - 37 T^{3} + 216 p T^{4} + p^{3} T^{6} \) | 3.79.a_ii_abl |
| 83 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 819 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ad_et_abfn |
| 89 | $A_4\times C_2$ | \( 1 - 3 T + 141 T^{2} - 477 T^{3} + 141 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ad_fl_asj |
| 97 | $A_4\times C_2$ | \( 1 + 27 T + 507 T^{2} + 5697 T^{3} + 507 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.bb_tn_ild |
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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
−9.287462173010484686242200106367, −8.390809754339466933084874926640, −8.151499876253415475194308274018, −8.087189605476788336123969203598, −7.49756203688192801529063262585, −7.43749929542951495063720060021, −7.18967206065112398106381009426, −6.70326199027002821085961793528, −6.66448133315458519249332646712, −6.58781025153773325652385679109, −5.91691072112951207048917525373, −5.83348067941653736669999364331, −5.73473718015344067586410559315, −5.00357626684990654874982756324, −4.89439261101250332280330375485, −4.86872774957665087235965293213, −4.35418322589260293864296790573, −4.12846236702990694789646112812, −3.90602582780873654651789929259, −3.49577844883699818740093502749, −3.07072327927044412072287395284, −3.00141236168871553933708662683, −1.97982037402250087835812747150, −1.97756921818019984034693284419, −1.78270532468388455931268464062, 0, 0, 0,
1.78270532468388455931268464062, 1.97756921818019984034693284419, 1.97982037402250087835812747150, 3.00141236168871553933708662683, 3.07072327927044412072287395284, 3.49577844883699818740093502749, 3.90602582780873654651789929259, 4.12846236702990694789646112812, 4.35418322589260293864296790573, 4.86872774957665087235965293213, 4.89439261101250332280330375485, 5.00357626684990654874982756324, 5.73473718015344067586410559315, 5.83348067941653736669999364331, 5.91691072112951207048917525373, 6.58781025153773325652385679109, 6.66448133315458519249332646712, 6.70326199027002821085961793528, 7.18967206065112398106381009426, 7.43749929542951495063720060021, 7.49756203688192801529063262585, 8.087189605476788336123969203598, 8.151499876253415475194308274018, 8.390809754339466933084874926640, 9.287462173010484686242200106367
