| L(s) = 1 | − 2-s − 2·4-s − 3·5-s + 2·8-s + 3·10-s − 4·11-s + 2·13-s + 16-s − 8·17-s − 3·19-s + 6·20-s + 4·22-s − 2·23-s + 6·25-s − 2·26-s − 14·29-s − 8·31-s + 32-s + 8·34-s + 8·37-s + 3·38-s − 6·40-s − 10·41-s + 2·43-s + 8·44-s + 2·46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 1.34·5-s + 0.707·8-s + 0.948·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.688·19-s + 1.34·20-s + 0.852·22-s − 0.417·23-s + 6/5·25-s − 0.392·26-s − 2.59·29-s − 1.43·31-s + 0.176·32-s + 1.37·34-s + 1.31·37-s + 0.486·38-s − 0.948·40-s − 1.56·41-s + 0.304·43-s + 1.20·44-s + 0.294·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.2.b_d_d |
| 7 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 2 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.7.a_r_ac |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 86 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_bj_di |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 62 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bb_ack |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 63 T^{2} + 276 T^{3} + 63 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.i_cl_kq |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 144 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_bl_fo |
| 29 | $S_4\times C_2$ | \( 1 + 14 T + 129 T^{2} + 822 T^{3} + 129 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.o_ez_bfq |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 69 T^{2} + 404 T^{3} + 69 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.i_cr_po |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 67 T^{2} - 566 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_cp_avu |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 81 T^{2} + 630 T^{3} + 81 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_dd_yg |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 98 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_dp_adu |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 125 T^{2} + 464 T^{3} + 125 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_ev_rw |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 235 T^{2} + 1788 T^{3} + 235 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.q_jb_cqu |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 13 T^{2} - 252 T^{3} + 13 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.k_n_ajs |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 87 T^{2} - 16 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_dj_aq |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 149 T^{2} - 84 T^{3} + 149 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ac_ft_adg |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 121 T^{2} + 84 T^{3} + 121 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.c_er_dg |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 139 T^{2} + 184 T^{3} + 139 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.e_fj_hc |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1496 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.m_gj_cfo |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 217 T^{2} - 384 T^{3} + 217 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ac_ij_aou |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 285 T^{2} + 1434 T^{3} + 285 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.i_kz_cde |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 243 T^{2} - 2414 T^{3} + 243 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ao_jj_adow |
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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
−9.393682515611858635033063989952, −9.250506927832235984376613238962, −8.838080552879132270965886573600, −8.733642264418716874223773661449, −8.254958039033611245947705308886, −8.137862052657435147089718735627, −7.960446986665253106922304295454, −7.54429937180210059777547235748, −7.46073624914392879272783671140, −6.92314766401111405480251776681, −6.66779493122786308787057327452, −6.23329852794249650966350817390, −6.16093632274777166886018763130, −5.54772461279799920237609475033, −5.17880757555621326773839027827, −5.01900368274190604972672753882, −4.51671389343548031183935496159, −4.24785638258284533029777644249, −4.19544875687752293345483636572, −3.53122431756953327878001245859, −3.37380310414299470778390920867, −2.91223550127613637702314987396, −2.37118975778481581088323203489, −1.75196341962709156271069675866, −1.51008341597215108771841449340, 0, 0, 0,
1.51008341597215108771841449340, 1.75196341962709156271069675866, 2.37118975778481581088323203489, 2.91223550127613637702314987396, 3.37380310414299470778390920867, 3.53122431756953327878001245859, 4.19544875687752293345483636572, 4.24785638258284533029777644249, 4.51671389343548031183935496159, 5.01900368274190604972672753882, 5.17880757555621326773839027827, 5.54772461279799920237609475033, 6.16093632274777166886018763130, 6.23329852794249650966350817390, 6.66779493122786308787057327452, 6.92314766401111405480251776681, 7.46073624914392879272783671140, 7.54429937180210059777547235748, 7.960446986665253106922304295454, 8.137862052657435147089718735627, 8.254958039033611245947705308886, 8.733642264418716874223773661449, 8.838080552879132270965886573600, 9.250506927832235984376613238962, 9.393682515611858635033063989952
