| L(s) = 1 | − 2·2-s − 3·5-s + 3·7-s + 2·8-s − 5·9-s + 6·10-s − 8·11-s − 6·14-s + 4·17-s + 10·18-s − 19-s + 16·22-s − 3·23-s − 5·25-s − 2·27-s − 7·29-s − 11·31-s − 8·34-s − 9·35-s − 12·37-s + 2·38-s − 6·40-s − 2·41-s + 13·43-s + 15·45-s + 6·46-s − 19·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.34·5-s + 1.13·7-s + 0.707·8-s − 5/3·9-s + 1.89·10-s − 2.41·11-s − 1.60·14-s + 0.970·17-s + 2.35·18-s − 0.229·19-s + 3.41·22-s − 0.625·23-s − 25-s − 0.384·27-s − 1.29·29-s − 1.97·31-s − 1.37·34-s − 1.52·35-s − 1.97·37-s + 0.324·38-s − 0.948·40-s − 0.312·41-s + 1.98·43-s + 2.23·45-s + 0.884·46-s − 2.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 7 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + p^{3} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_e_g |
| 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.3.a_f_c |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 14 T^{2} + 29 T^{3} + 14 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_o_bd |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 51 T^{2} + 186 T^{3} + 51 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.i_bz_he |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} - 6 p T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_br_ady |
| 19 | $S_4\times C_2$ | \( 1 + T - 2 T^{2} - 155 T^{3} - 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.19.b_ac_afz |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 44 T^{2} + 59 T^{3} + 44 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.d_bs_ch |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 66 T^{2} + 411 T^{3} + 66 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.h_co_pv |
| 31 | $S_4\times C_2$ | \( 1 + 11 T + 112 T^{2} + 617 T^{3} + 112 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.l_ei_xt |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 129 T^{2} + 834 T^{3} + 129 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.m_ez_bgc |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 71 T^{2} + 124 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.c_ct_eu |
| 43 | $S_4\times C_2$ | \( 1 - 13 T + 164 T^{2} - 1101 T^{3} + 164 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.an_gi_abqj |
| 47 | $S_4\times C_2$ | \( 1 + 19 T + 246 T^{2} + 1923 T^{3} + 246 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.t_jm_cvz |
| 53 | $S_4\times C_2$ | \( 1 - T + 150 T^{2} - 93 T^{3} + 150 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ab_fu_adp |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 145 T^{2} + 288 T^{3} + 145 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_fp_lc |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 211 T^{2} - 1556 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_id_achw |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 233 T^{2} - 1592 T^{3} + 233 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_iz_acjg |
| 71 | $S_4\times C_2$ | \( 1 + 14 T + 159 T^{2} + 1098 T^{3} + 159 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.o_gd_bqg |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 128 T^{2} - 1345 T^{3} + 128 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_ey_abzt |
| 79 | $S_4\times C_2$ | \( 1 - 13 T + 200 T^{2} - 1869 T^{3} + 200 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.an_hs_actx |
| 83 | $S_4\times C_2$ | \( 1 + T + 136 T^{2} + 3 T^{3} + 136 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.83.b_fg_d |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 212 T^{2} + 307 T^{3} + 212 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.d_ie_lv |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 116 T^{2} + 727 T^{3} + 116 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.p_em_bbz |
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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
−9.013678130512164891924808807312, −8.732155673504000960665063683088, −8.369901373628737910011274279396, −8.258717088927275178467237142385, −8.161240540532178346467687809835, −7.80245891872403073454714680276, −7.69182664678827542434551902619, −7.46799909508865815190875577757, −7.23095812059792818201357603362, −6.69962179507249985272396171165, −6.28791414660476462030742661463, −5.73081306614458360657314570446, −5.65690390623454708924253425086, −5.32217950573224615090373766316, −5.17505644066725341479920994358, −5.03269879808719851186930675136, −4.28602982382738700187627583875, −4.09851570007425064277115492293, −3.66676379338309043376831998345, −3.30005171903209262990838650036, −3.25177630585061785739351572903, −2.41985883830778310917852833388, −2.17138339161372740952639961251, −1.88066017648591091058258502488, −1.05900729119518398528258095041, 0, 0, 0,
1.05900729119518398528258095041, 1.88066017648591091058258502488, 2.17138339161372740952639961251, 2.41985883830778310917852833388, 3.25177630585061785739351572903, 3.30005171903209262990838650036, 3.66676379338309043376831998345, 4.09851570007425064277115492293, 4.28602982382738700187627583875, 5.03269879808719851186930675136, 5.17505644066725341479920994358, 5.32217950573224615090373766316, 5.65690390623454708924253425086, 5.73081306614458360657314570446, 6.28791414660476462030742661463, 6.69962179507249985272396171165, 7.23095812059792818201357603362, 7.46799909508865815190875577757, 7.69182664678827542434551902619, 7.80245891872403073454714680276, 8.161240540532178346467687809835, 8.258717088927275178467237142385, 8.369901373628737910011274279396, 8.732155673504000960665063683088, 9.013678130512164891924808807312
