| L(s) = 1 | − 4·2-s + 7·4-s − 7·8-s − 11-s + 5·13-s + 7·16-s − 2·17-s − 3·19-s + 4·22-s − 8·23-s − 20·26-s + 7·29-s − 5·31-s − 14·32-s + 8·34-s + 5·37-s + 12·38-s − 41-s + 5·43-s − 7·44-s + 32·46-s − 3·47-s − 14·49-s + 35·52-s − 19·53-s − 28·58-s − 10·59-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 7/2·4-s − 2.47·8-s − 0.301·11-s + 1.38·13-s + 7/4·16-s − 0.485·17-s − 0.688·19-s + 0.852·22-s − 1.66·23-s − 3.92·26-s + 1.29·29-s − 0.898·31-s − 2.47·32-s + 1.37·34-s + 0.821·37-s + 1.94·38-s − 0.156·41-s + 0.762·43-s − 1.05·44-s + 4.71·46-s − 0.437·47-s − 2·49-s + 4.85·52-s − 2.60·53-s − 3.67·58-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $C_6$ | \( 1 + p^{2} T + 9 T^{2} + 15 T^{3} + 9 p T^{4} + p^{4} T^{5} + p^{3} T^{6} \) | 3.2.e_j_p |
| 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) | 3.7.a_o_h |
| 11 | $A_4\times C_2$ | \( 1 + T + 17 T^{2} + 35 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.11.b_r_bj |
| 13 | $A_4\times C_2$ | \( 1 - 5 T + 45 T^{2} - 131 T^{3} + 45 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.af_bt_afb |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 36 T^{2} + 81 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_bk_dd |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 88 T^{2} + 381 T^{3} + 88 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.i_dk_or |
| 29 | $A_4\times C_2$ | \( 1 - 7 T + 45 T^{2} - 315 T^{3} + 45 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ah_bt_amd |
| 31 | $A_4\times C_2$ | \( 1 + 5 T + 57 T^{2} + 353 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.f_cf_np |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + 103 T^{2} - 371 T^{3} + 103 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.af_dz_aoh |
| 41 | $A_4\times C_2$ | \( 1 + T + 9 T^{2} - 339 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.41.b_j_anb |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 72 T^{2} - 137 T^{3} + 72 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.af_cu_afh |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 137 T^{2} + 269 T^{3} + 137 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.d_fh_kj |
| 53 | $A_4\times C_2$ | \( 1 + 19 T + 214 T^{2} + 1707 T^{3} + 214 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.t_ig_cnr |
| 59 | $A_4\times C_2$ | \( 1 + 10 T + 145 T^{2} + 852 T^{3} + 145 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.k_fp_bgu |
| 61 | $A_4\times C_2$ | \( 1 + 17 T + 249 T^{2} + 2033 T^{3} + 249 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.r_jp_daf |
| 67 | $A_4\times C_2$ | \( 1 + T + 59 T^{2} + 693 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.67.b_ch_bar |
| 71 | $A_4\times C_2$ | \( 1 - 19 T + 268 T^{2} - 2391 T^{3} + 268 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.at_ki_adnz |
| 73 | $A_4\times C_2$ | \( 1 + T + 49 T^{2} - 317 T^{3} + 49 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.73.b_bx_amf |
| 79 | $A_4\times C_2$ | \( 1 - 18 T + 324 T^{2} - 2941 T^{3} + 324 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.as_mm_aejd |
| 83 | $A_4\times C_2$ | \( 1 + 13 T + 247 T^{2} + 2019 T^{3} + 247 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.n_jn_czr |
| 89 | $A_4\times C_2$ | \( 1 + 2 T + 168 T^{2} + 75 T^{3} + 168 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.c_gm_cx |
| 97 | $A_4\times C_2$ | \( 1 - 5 T + 297 T^{2} - 971 T^{3} + 297 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.af_ll_ablj |
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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
−8.100984142282920194712959234942, −7.71704200896829465038019584159, −7.46266908233986228205309480545, −7.46082487381810938075252772941, −6.76209301162401779026100789245, −6.59909169108723575692195935401, −6.38823138085966592611306674317, −6.36085002128734901545080700788, −5.86676251130974439579647131234, −5.82015000606447604151477099607, −5.51127692085867322491144630236, −4.85016213006715481697956828238, −4.84629155472430061537404777657, −4.53161878922512378318427726451, −4.30285423089235161442316339831, −3.69388816883184162113157474541, −3.64630473049700540338608358491, −3.21261600856183500925255524688, −3.17195507606839336942758573420, −2.43657219240356756124696537364, −2.23414646177415298025787896359, −2.03241682376835517154801470204, −1.35707431756539852706439668830, −1.18425668638879174901563770241, −1.15498492282229143568181371363, 0, 0, 0,
1.15498492282229143568181371363, 1.18425668638879174901563770241, 1.35707431756539852706439668830, 2.03241682376835517154801470204, 2.23414646177415298025787896359, 2.43657219240356756124696537364, 3.17195507606839336942758573420, 3.21261600856183500925255524688, 3.64630473049700540338608358491, 3.69388816883184162113157474541, 4.30285423089235161442316339831, 4.53161878922512378318427726451, 4.84629155472430061537404777657, 4.85016213006715481697956828238, 5.51127692085867322491144630236, 5.82015000606447604151477099607, 5.86676251130974439579647131234, 6.36085002128734901545080700788, 6.38823138085966592611306674317, 6.59909169108723575692195935401, 6.76209301162401779026100789245, 7.46082487381810938075252772941, 7.46266908233986228205309480545, 7.71704200896829465038019584159, 8.100984142282920194712959234942
