| L(s) = 1 | − 3·3-s + 5-s − 3·7-s + 9-s − 6·11-s + 4·13-s − 3·15-s − 3·17-s − 2·19-s + 9·21-s − 6·23-s − 9·25-s + 7·27-s − 4·29-s − 13·31-s + 18·33-s − 3·35-s − 12·39-s − 5·41-s − 5·43-s + 45-s − 6·47-s + 6·49-s + 9·51-s − 9·53-s − 6·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.774·15-s − 0.727·17-s − 0.458·19-s + 1.96·21-s − 1.25·23-s − 9/5·25-s + 1.34·27-s − 0.742·29-s − 2.33·31-s + 3.13·33-s − 0.507·35-s − 1.92·39-s − 0.780·41-s − 0.762·43-s + 0.149·45-s − 0.875·47-s + 6/7·49-s + 1.26·51-s − 1.23·53-s − 0.809·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 8 T^{2} + 14 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_i_o |
| 5 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 12 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ab_k_am |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.11.g_bt_fk |
| 13 | $D_{6}$ | \( 1 - 4 T + 19 T^{2} - 88 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_t_adk |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 44 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.c_bh_bs |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 65 T^{2} + 244 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cn_jk |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 43 T^{2} + 264 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.e_br_ke |
| 31 | $S_4\times C_2$ | \( 1 + 13 T + 134 T^{2} + 798 T^{3} + 134 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.n_fe_bes |
| 37 | $S_4\times C_2$ | \( 1 + 47 T^{2} - 64 T^{3} + 47 p T^{4} + p^{3} T^{6} \) | 3.37.a_bv_acm |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 54 T^{2} + 24 T^{3} + 54 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.f_cc_y |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 98 T^{2} + 282 T^{3} + 98 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_du_kw |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 137 T^{2} + 532 T^{3} + 137 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_fh_um |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 104 T^{2} + 628 T^{3} + 104 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_ea_ye |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 217 T^{2} + 1620 T^{3} + 217 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.o_ij_cki |
| 61 | $S_4\times C_2$ | \( 1 - 9 T + 206 T^{2} - 1112 T^{3} + 206 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.aj_hy_abqu |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 120 T^{2} + 1150 T^{3} + 120 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.n_eq_bsg |
| 71 | $S_4\times C_2$ | \( 1 + 24 T + 341 T^{2} + 3472 T^{3} + 341 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.y_nd_fdo |
| 73 | $S_4\times C_2$ | \( 1 + 7 T + 78 T^{2} + 304 T^{3} + 78 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.h_da_ls |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 452 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.g_eb_rk |
| 83 | $S_4\times C_2$ | \( 1 + 65 T^{2} - 424 T^{3} + 65 p T^{4} + p^{3} T^{6} \) | 3.83.a_cn_aqi |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 227 T^{2} - 1096 T^{3} + 227 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ai_it_abqe |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 58 T^{2} + 364 T^{3} + 58 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.af_cg_oa |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.397188602376956882211462145652, −8.975221618698435545434610028405, −8.824416743148549493540845240328, −8.795303434198117177399428466590, −7.955210608007411579953787605234, −7.904365496951229484146703267841, −7.80788700663037247956075969132, −7.36747990910192031476062282158, −6.82448387374734144699018174841, −6.74937230890652095031384090996, −6.15878612426621215557014410536, −6.09937217224428049505955913252, −5.98958117270713965028109659456, −5.53605431312127747713034786267, −5.41762592723919438993900673887, −5.25818088986064235497135796336, −4.55261187958238859049766801837, −4.39107868653115196038367174544, −3.97339911725815955559123416548, −3.40290271568107814091269375596, −3.23449356450603061001265517615, −2.90152227871142188849289545674, −2.11611549770330803829537994729, −1.97112939376761476912608640322, −1.46883846965923117130653728435, 0, 0, 0,
1.46883846965923117130653728435, 1.97112939376761476912608640322, 2.11611549770330803829537994729, 2.90152227871142188849289545674, 3.23449356450603061001265517615, 3.40290271568107814091269375596, 3.97339911725815955559123416548, 4.39107868653115196038367174544, 4.55261187958238859049766801837, 5.25818088986064235497135796336, 5.41762592723919438993900673887, 5.53605431312127747713034786267, 5.98958117270713965028109659456, 6.09937217224428049505955913252, 6.15878612426621215557014410536, 6.74937230890652095031384090996, 6.82448387374734144699018174841, 7.36747990910192031476062282158, 7.80788700663037247956075969132, 7.904365496951229484146703267841, 7.955210608007411579953787605234, 8.795303434198117177399428466590, 8.824416743148549493540845240328, 8.975221618698435545434610028405, 9.397188602376956882211462145652
