| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s + 9·6-s − 2·7-s − 4·8-s + 6·9-s + 2·11-s − 12·12-s + 6·13-s + 6·14-s + 3·16-s − 4·17-s − 18·18-s − 8·19-s + 6·21-s − 6·22-s − 14·23-s + 12·24-s − 18·26-s − 10·27-s − 8·28-s + 16·29-s − 3·31-s + 32-s − 6·33-s + 12·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s + 3.67·6-s − 0.755·7-s − 1.41·8-s + 2·9-s + 0.603·11-s − 3.46·12-s + 1.66·13-s + 1.60·14-s + 3/4·16-s − 0.970·17-s − 4.24·18-s − 1.83·19-s + 1.30·21-s − 1.27·22-s − 2.91·23-s + 2.44·24-s − 3.53·26-s − 1.92·27-s − 1.51·28-s + 2.97·29-s − 0.538·31-s + 0.176·32-s − 1.04·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.2.d_f_h |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 38 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_j_bm |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 36 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_v_abk |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 154 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ag_bv_afy |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 47 T^{2} + 116 T^{3} + 47 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_bv_em |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 272 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_cf_km |
| 23 | $S_4\times C_2$ | \( 1 + 14 T + 129 T^{2} + 720 T^{3} + 129 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.o_ez_bbs |
| 29 | $S_4\times C_2$ | \( 1 - 16 T + 149 T^{2} - 938 T^{3} + 149 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.aq_ft_abkc |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 14 p T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_eh_aty |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 344 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ae_bj_ang |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} + 28 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_cr_bc |
| 47 | $S_4\times C_2$ | \( 1 + 14 T + 49 T^{2} - 72 T^{3} + 49 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.o_bx_acu |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 87 T^{2} + 948 T^{3} + 87 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.i_dj_bkm |
| 59 | $S_4\times C_2$ | \( 1 + 26 T + 379 T^{2} + 3534 T^{3} + 379 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ba_op_ffy |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 227 T^{2} + 1900 T^{3} + 227 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.s_it_cvc |
| 67 | $S_4\times C_2$ | \( 1 + 65 T^{2} + 274 T^{3} + 65 p T^{4} + p^{3} T^{6} \) | 3.67.a_cn_ko |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 215 T^{2} - 566 T^{3} + 215 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ae_ih_avu |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 155 T^{2} - 1618 T^{3} + 155 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_fz_ackg |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} - 1132 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_dp_abro |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 205 T^{2} - 1272 T^{3} + 205 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ak_hx_abwy |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 249 T^{2} + 1018 T^{3} + 249 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_jp_bne |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 51 T^{2} - 160 T^{3} + 51 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.i_bz_age |
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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
−8.443109102787469701403919676749, −8.028378500570744554438195769805, −7.978690815167482388694833811485, −7.75877237126375300651777820893, −7.61083425112795703088988284003, −6.79787926386540506756196034120, −6.60016014154775862484028723995, −6.56978481264360987836007181278, −6.29504161238328345576163224265, −6.26075432874500442319120208006, −6.02037895175382806108541578563, −5.85441261264160422923414388709, −5.15503122332255557062415668073, −4.75551778263410917279966215282, −4.75538515883274546015282687339, −4.28846284634197762271215964431, −3.95750253573367793618815874551, −3.92183914233279468775818914833, −3.35102369360875437835067080538, −2.93660955594812261485886169952, −2.46419353973355744717973158392, −2.15262907399528912240254522049, −1.61374315381691948804758067129, −1.25773606251329176780813340850, −1.08366372735667782461405333620, 0, 0, 0,
1.08366372735667782461405333620, 1.25773606251329176780813340850, 1.61374315381691948804758067129, 2.15262907399528912240254522049, 2.46419353973355744717973158392, 2.93660955594812261485886169952, 3.35102369360875437835067080538, 3.92183914233279468775818914833, 3.95750253573367793618815874551, 4.28846284634197762271215964431, 4.75538515883274546015282687339, 4.75551778263410917279966215282, 5.15503122332255557062415668073, 5.85441261264160422923414388709, 6.02037895175382806108541578563, 6.26075432874500442319120208006, 6.29504161238328345576163224265, 6.56978481264360987836007181278, 6.60016014154775862484028723995, 6.79787926386540506756196034120, 7.61083425112795703088988284003, 7.75877237126375300651777820893, 7.978690815167482388694833811485, 8.028378500570744554438195769805, 8.443109102787469701403919676749
