| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s − 3·5-s + 9·6-s + 2·7-s + 4·8-s + 6·9-s − 9·10-s + 2·11-s + 12·12-s − 6·13-s + 6·14-s − 9·15-s + 3·16-s + 4·17-s + 18·18-s − 8·19-s − 12·20-s + 6·21-s + 6·22-s + 14·23-s + 12·24-s + 6·25-s − 18·26-s + 10·27-s + 8·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s + 0.755·7-s + 1.41·8-s + 2·9-s − 2.84·10-s + 0.603·11-s + 3.46·12-s − 1.66·13-s + 1.60·14-s − 2.32·15-s + 3/4·16-s + 0.970·17-s + 4.24·18-s − 1.83·19-s − 2.68·20-s + 1.30·21-s + 1.27·22-s + 2.91·23-s + 2.44·24-s + 6/5·25-s − 3.53·26-s + 1.92·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.92607336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.92607336\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 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.ad_f_ah |
| 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.ac_j_abm |
| 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.g_bv_fy |
| 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.ae_bv_aem |
| 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.ao_ez_abbs |
| 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.i_eh_ty |
| 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.c_cr_abc |
| 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.ao_bx_cu |
| 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.ai_dj_abkm |
| 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_ako |
| 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.m_fz_ckg |
| 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.k_hx_bwy |
| 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.ai_bz_ge |
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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
−10.02670981727335839668810668426, −9.325087241850165629395985636598, −9.095691646859950272365660334443, −8.899858082483501517835042730927, −8.550826196850240729867402955543, −8.449098712253324983732910114815, −7.85809214604848448497196038107, −7.56790828626573431443370816449, −7.36363329354494068992322168700, −7.17997498153912541963279917808, −6.83082068114520013645501040424, −6.23855580122212105384265967024, −6.20872296595052740990885103658, −5.24778498026556106592030325918, −5.03027204279059755726977174379, −4.78081528419228883280157842509, −4.48897657483099978241596569676, −4.28186349879091723733665393629, −4.06029850489067973731501129336, −3.29561464959973720682366888093, −3.11840522466367847227150291557, −2.94821973247290220612455236669, −2.36609147327689149439071684279, −1.76927804132056196680464765858, −1.03613176680905386854326379841,
1.03613176680905386854326379841, 1.76927804132056196680464765858, 2.36609147327689149439071684279, 2.94821973247290220612455236669, 3.11840522466367847227150291557, 3.29561464959973720682366888093, 4.06029850489067973731501129336, 4.28186349879091723733665393629, 4.48897657483099978241596569676, 4.78081528419228883280157842509, 5.03027204279059755726977174379, 5.24778498026556106592030325918, 6.20872296595052740990885103658, 6.23855580122212105384265967024, 6.83082068114520013645501040424, 7.17997498153912541963279917808, 7.36363329354494068992322168700, 7.56790828626573431443370816449, 7.85809214604848448497196038107, 8.449098712253324983732910114815, 8.550826196850240729867402955543, 8.899858082483501517835042730927, 9.095691646859950272365660334443, 9.325087241850165629395985636598, 10.02670981727335839668810668426
