| L(s) = 1 | + 3·3-s − 3·4-s − 3·5-s − 3·7-s − 8-s + 6·9-s − 3·11-s − 9·12-s − 3·13-s − 9·15-s + 3·16-s − 9·17-s + 9·20-s − 9·21-s + 6·23-s − 3·24-s − 6·25-s + 10·27-s + 9·28-s − 9·29-s − 24·31-s + 6·32-s − 9·33-s + 9·35-s − 18·36-s − 6·37-s − 9·39-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 3/2·4-s − 1.34·5-s − 1.13·7-s − 0.353·8-s + 2·9-s − 0.904·11-s − 2.59·12-s − 0.832·13-s − 2.32·15-s + 3/4·16-s − 2.18·17-s + 2.01·20-s − 1.96·21-s + 1.25·23-s − 0.612·24-s − 6/5·25-s + 1.92·27-s + 1.70·28-s − 1.67·29-s − 4.31·31-s + 1.06·32-s − 1.56·33-s + 1.52·35-s − 3·36-s − 0.986·37-s − 1.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | | \( 1 \) | |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T^{2} + T^{3} + 3 p T^{4} + p^{3} T^{6} \) | 3.2.a_d_b |
| 5 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_p_bb |
| 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 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 29 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_p_bd |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 30 T^{2} + 59 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.d_be_ch |
| 17 | $A_4\times C_2$ | \( 1 + 9 T + 57 T^{2} + 253 T^{3} + 57 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.j_cf_jt |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 54 T^{2} - 203 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_cc_ahv |
| 29 | $A_4\times C_2$ | \( 1 + 9 T + 66 T^{2} + 469 T^{3} + 66 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.j_co_sb |
| 31 | $A_4\times C_2$ | \( 1 + 24 T + 282 T^{2} + 1977 T^{3} + 282 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.y_kw_cyb |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 102 T^{2} + 393 T^{3} + 102 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_dy_pd |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 441 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ds_aqz |
| 43 | $A_4\times C_2$ | \( 1 + 21 T + 240 T^{2} + 1825 T^{3} + 240 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.v_jg_csf |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 495 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.d_bh_tb |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 210 T^{2} - 1619 T^{3} + 210 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.as_ic_ackh |
| 59 | $A_4\times C_2$ | \( 1 - 15 T + 243 T^{2} - 1859 T^{3} + 243 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ap_jj_actn |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 1135 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.j_en_brr |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 508 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_ez_ato |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 204 T^{2} - 1125 T^{3} + 204 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_hw_abrh |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 767 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_gm_bdn |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 216 T^{2} + 1369 T^{3} + 216 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.j_ii_car |
| 83 | $A_4\times C_2$ | \( 1 + 15 T + 285 T^{2} + 2439 T^{3} + 285 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.p_kz_dpv |
| 89 | $A_4\times C_2$ | \( 1 + 42 T^{2} + 1125 T^{3} + 42 p T^{4} + p^{3} T^{6} \) | 3.89.a_bq_brh |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 246 T^{2} - 895 T^{3} + 246 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ag_jm_abil |
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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
−9.323809592767699733041422884728, −8.804554995447597577420281045048, −8.715343159726752698457853086915, −8.534461523800480716849337323202, −8.140729276835605694161798542535, −7.932988318545102605537739839666, −7.58957700264625029367251607238, −7.24014876292282001736558137158, −7.04525855571811039392869532583, −6.98265403513182835066762774541, −6.59289796508453320225512635831, −6.03299243916312091609718484813, −5.63841317741986418633698158685, −5.21405868734692212836788368330, −5.00539491795488079914459062827, −4.90029569959468801424097765184, −4.13807983145093895664224226104, −3.92973365701156412986090157253, −3.92782003901918499052843945430, −3.54058177996859447645360824686, −3.28294613342035413529136156420, −2.76558405894223687654421783072, −2.42661179110749764027206671404, −1.90681417130665009428804541336, −1.67527569965281031494178029870, 0, 0, 0,
1.67527569965281031494178029870, 1.90681417130665009428804541336, 2.42661179110749764027206671404, 2.76558405894223687654421783072, 3.28294613342035413529136156420, 3.54058177996859447645360824686, 3.92782003901918499052843945430, 3.92973365701156412986090157253, 4.13807983145093895664224226104, 4.90029569959468801424097765184, 5.00539491795488079914459062827, 5.21405868734692212836788368330, 5.63841317741986418633698158685, 6.03299243916312091609718484813, 6.59289796508453320225512635831, 6.98265403513182835066762774541, 7.04525855571811039392869532583, 7.24014876292282001736558137158, 7.58957700264625029367251607238, 7.932988318545102605537739839666, 8.140729276835605694161798542535, 8.534461523800480716849337323202, 8.715343159726752698457853086915, 8.804554995447597577420281045048, 9.323809592767699733041422884728
