| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 3·5-s − 6·6-s − 4·7-s − 4·8-s + 6·9-s + 6·10-s + 5·11-s + 9·12-s + 2·13-s + 8·14-s − 9·15-s + 3·16-s − 12·18-s − 5·19-s − 9·20-s − 12·21-s − 10·22-s − 4·23-s − 12·24-s + 6·25-s − 4·26-s + 10·27-s − 12·28-s + 29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s − 2.44·6-s − 1.51·7-s − 1.41·8-s + 2·9-s + 1.89·10-s + 1.50·11-s + 2.59·12-s + 0.554·13-s + 2.13·14-s − 2.32·15-s + 3/4·16-s − 2.82·18-s − 1.14·19-s − 2.01·20-s − 2.61·21-s − 2.13·22-s − 0.834·23-s − 2.44·24-s + 6/5·25-s − 0.784·26-s + 1.92·27-s − 2.26·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 17 | | \( 1 \) | |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_b_a |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 18 T^{2} + 46 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_s_bu |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 17 T^{2} - 46 T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.af_r_abu |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 32 T^{2} - 48 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bg_abw |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 32 T^{2} + 129 T^{3} + 32 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.f_bg_ez |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 152 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_bp_fw |
| 29 | $S_4\times C_2$ | \( 1 - T + 79 T^{2} - 54 T^{3} + 79 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ab_db_acc |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 86 T^{2} + 396 T^{3} + 86 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.o_di_pg |
| 37 | $S_4\times C_2$ | \( 1 + 28 T + 364 T^{2} + 2806 T^{3} + 364 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.bc_oa_edy |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 83 T^{2} + 262 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.d_df_kc |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 178 T^{3} + 6 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_g_gw |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 388 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_cn_oy |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 131 T^{2} - 628 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_fb_aye |
| 59 | $S_4\times C_2$ | \( 1 + 13 T + 225 T^{2} + 1578 T^{3} + 225 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.n_ir_cis |
| 61 | $S_4\times C_2$ | \( 1 + 13 T + 206 T^{2} + 1533 T^{3} + 206 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.n_hy_cgz |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 86 T^{2} + 318 T^{3} + 86 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.e_di_mg |
| 71 | $S_4\times C_2$ | \( 1 - T + 85 T^{2} - 102 T^{3} + 85 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ab_dh_ady |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) | 3.73.s_mp_efk |
| 79 | $S_4\times C_2$ | \( 1 - 7 T + 229 T^{2} - 1090 T^{3} + 229 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ah_iv_abpy |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 193 T^{2} + 2340 T^{3} + 193 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.o_hl_dma |
| 89 | $S_4\times C_2$ | \( 1 - 13 T + 251 T^{2} - 2318 T^{3} + 251 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.an_jr_adle |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 176 T^{2} - 558 T^{3} + 176 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ae_gu_avm |
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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
−7.66542392823415937106863354602, −7.63837179050755192291584772145, −7.32212774148828181250019082570, −7.19168644190897411685010507038, −6.88609319508851068512407141506, −6.86743560610702567539461323114, −6.31828148262948789199008901710, −6.28217725834550399158089454395, −6.19985155265478094622012295789, −5.69629602740752934891772845303, −5.23706183960893383272532141706, −4.89605199118641276326402485478, −4.78046858929699760518925540880, −4.15859339033917740078945275982, −4.03570842446857934567763426696, −3.71006469171772982494385485451, −3.43909616810300534561031163200, −3.43352515266709399358820435574, −3.26324980153055200622670045824, −2.84274501204148929115631884852, −2.34995638607595177801274129476, −1.95507726621922571373619154401, −1.72025300215463729482018405047, −1.57443709912931600698831119615, −1.14341103837742123392013743357, 0, 0, 0,
1.14341103837742123392013743357, 1.57443709912931600698831119615, 1.72025300215463729482018405047, 1.95507726621922571373619154401, 2.34995638607595177801274129476, 2.84274501204148929115631884852, 3.26324980153055200622670045824, 3.43352515266709399358820435574, 3.43909616810300534561031163200, 3.71006469171772982494385485451, 4.03570842446857934567763426696, 4.15859339033917740078945275982, 4.78046858929699760518925540880, 4.89605199118641276326402485478, 5.23706183960893383272532141706, 5.69629602740752934891772845303, 6.19985155265478094622012295789, 6.28217725834550399158089454395, 6.31828148262948789199008901710, 6.86743560610702567539461323114, 6.88609319508851068512407141506, 7.19168644190897411685010507038, 7.32212774148828181250019082570, 7.63837179050755192291584772145, 7.66542392823415937106863354602
