| L(s) = 1 | − 3·3-s + 3·5-s + 7·7-s + 9-s + 2·11-s + 9·13-s − 9·15-s + 17-s + 6·19-s − 21·21-s − 3·23-s + 6·25-s + 5·27-s − 3·29-s − 9·31-s − 6·33-s + 21·35-s − 8·37-s − 27·39-s + 8·41-s + 7·43-s + 3·45-s + 14·47-s + 27·49-s − 3·51-s + 11·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 2.64·7-s + 1/3·9-s + 0.603·11-s + 2.49·13-s − 2.32·15-s + 0.242·17-s + 1.37·19-s − 4.58·21-s − 0.625·23-s + 6/5·25-s + 0.962·27-s − 0.557·29-s − 1.61·31-s − 1.04·33-s + 3.54·35-s − 1.31·37-s − 4.32·39-s + 1.24·41-s + 1.06·43-s + 0.447·45-s + 2.04·47-s + 27/7·49-s − 0.420·51-s + 1.51·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.570790883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.570790883\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + p T + 8 T^{2} + 16 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_i_q |
| 7 | $S_4\times C_2$ | \( 1 - p T + 22 T^{2} - 52 T^{3} + 22 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.7.ah_w_aca |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 60 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_n_aci |
| 13 | $S_4\times C_2$ | \( 1 - 9 T + 62 T^{2} - 248 T^{3} + 62 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.aj_ck_ajo |
| 17 | $S_4\times C_2$ | \( 1 - T + 46 T^{2} - 36 T^{3} + 46 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ab_bu_abk |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 196 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ag_cb_aho |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 68 T^{2} + 136 T^{3} + 68 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.d_cq_fg |
| 31 | $S_4\times C_2$ | \( 1 + 9 T + 74 T^{2} + 394 T^{3} + 74 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.j_cw_pe |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 71 T^{2} + 264 T^{3} + 71 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_ct_ke |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 648 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ai_et_ayy |
| 43 | $S_4\times C_2$ | \( 1 - 7 T + 118 T^{2} - 496 T^{3} + 118 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ah_eo_atc |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 157 T^{2} - 1092 T^{3} + 157 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ao_gb_abqa |
| 53 | $S_4\times C_2$ | \( 1 - 11 T + 118 T^{2} - 1164 T^{3} + 118 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.al_eo_absu |
| 59 | $S_4\times C_2$ | \( 1 + T + 20 T^{2} - 298 T^{3} + 20 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.59.b_u_alm |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 108 T^{2} - 952 T^{3} + 108 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ah_ee_abkq |
| 67 | $S_4\times C_2$ | \( 1 + 26 T + 377 T^{2} + 3612 T^{3} + 377 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ba_on_fiy |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 149 T^{2} + 688 T^{3} + 149 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.i_ft_bam |
| 73 | $S_4\times C_2$ | \( 1 + 7 T + 98 T^{2} + 636 T^{3} + 98 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.h_du_ym |
| 79 | $S_4\times C_2$ | \( 1 - 17 T + 248 T^{2} - 2310 T^{3} + 248 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ar_jo_adkw |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 233 T^{2} + 608 T^{3} + 233 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.e_iz_xk |
| 89 | $S_4\times C_2$ | \( 1 - 32 T + 531 T^{2} - 5880 T^{3} + 531 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.abg_ul_aise |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 2 p T^{2} - 608 T^{3} + 2 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ad_hm_axk |
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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
−6.99312122688500227784839061433, −6.11534325829491636194950233276, −6.09215895397005197770957426040, −6.04297666535317162208386116981, −5.93148097457874319783041826294, −5.77247467831941747962120254346, −5.60962049609555931208141297852, −5.09561019640374383205508896911, −5.02885478581379776055576545508, −5.01940514423501809278177650958, −4.63621673675041854067697902098, −4.27859687531336200828676842915, −3.96537617986253440658057078914, −3.76321185027808865106244288211, −3.57782705051627347915476822492, −3.32778426813076204481671823141, −2.78598841364027109389648916998, −2.50943507619880391919095055356, −2.15438189738726794609489594648, −1.82100689880879603491686523410, −1.66887203458437798608674962233, −1.42602160766986621155226937223, −1.04962022796950107608160580480, −0.70436808992103657025096850303, −0.58688685318679662660595777148,
0.58688685318679662660595777148, 0.70436808992103657025096850303, 1.04962022796950107608160580480, 1.42602160766986621155226937223, 1.66887203458437798608674962233, 1.82100689880879603491686523410, 2.15438189738726794609489594648, 2.50943507619880391919095055356, 2.78598841364027109389648916998, 3.32778426813076204481671823141, 3.57782705051627347915476822492, 3.76321185027808865106244288211, 3.96537617986253440658057078914, 4.27859687531336200828676842915, 4.63621673675041854067697902098, 5.01940514423501809278177650958, 5.02885478581379776055576545508, 5.09561019640374383205508896911, 5.60962049609555931208141297852, 5.77247467831941747962120254346, 5.93148097457874319783041826294, 6.04297666535317162208386116981, 6.09215895397005197770957426040, 6.11534325829491636194950233276, 6.99312122688500227784839061433
