| L(s) = 1 | − 5·2-s − 3-s + 12·4-s − 4·5-s + 5·6-s − 20·8-s + 20·10-s − 6·11-s − 12·12-s − 4·13-s + 4·15-s + 30·16-s − 10·17-s − 4·19-s − 48·20-s + 30·22-s + 2·23-s + 20·24-s + 10·25-s + 20·26-s − 10·29-s − 20·30-s + 31-s − 45·32-s + 6·33-s + 50·34-s − 26·37-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 0.577·3-s + 6·4-s − 1.78·5-s + 2.04·6-s − 7.07·8-s + 6.32·10-s − 1.80·11-s − 3.46·12-s − 1.10·13-s + 1.03·15-s + 15/2·16-s − 2.42·17-s − 0.917·19-s − 10.7·20-s + 6.39·22-s + 0.417·23-s + 4.08·24-s + 2·25-s + 3.92·26-s − 1.85·29-s − 3.65·30-s + 0.179·31-s − 7.95·32-s + 1.04·33-s + 8.57·34-s − 4.27·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 31 | $C_4$ | \( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} \) | |
| good | 2 | $C_4\times C_2$ | \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.2.f_n_z_bn |
| 7 | $C_2^2:C_4$ | \( 1 + 3 T^{2} - 10 T^{3} + 39 T^{4} - 10 p T^{5} + 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_d_ak_bn |
| 11 | $C_2^2:C_4$ | \( 1 + 6 T + 5 T^{2} - 6 T^{3} + 49 T^{4} - 6 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.g_f_ag_bx |
| 13 | $C_2^2:C_4$ | \( 1 + 4 T - 7 T^{2} - 40 T^{3} + T^{4} - 40 p T^{5} - 7 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.e_ah_abo_b |
| 17 | $C_2^2:C_4$ | \( 1 + 10 T + 23 T^{2} - 10 p T^{3} - 1351 T^{4} - 10 p^{2} T^{5} + 23 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.k_x_ago_abzz |
| 19 | $C_4\times C_2$ | \( 1 + 4 T + 27 T^{2} + 122 T^{3} + 275 T^{4} + 122 p T^{5} + 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.e_bb_es_kp |
| 23 | $C_4\times C_2$ | \( 1 - 2 T - 19 T^{2} + 84 T^{3} + 269 T^{4} + 84 p T^{5} - 19 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ac_at_dg_kj |
| 29 | $C_2^2:C_4$ | \( 1 + 10 T + 11 T^{2} - 10 p T^{3} - 2239 T^{4} - 10 p^{2} T^{5} + 11 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.k_l_ale_adid |
| 37 | $D_{4}$ | \( ( 1 + 13 T + 115 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ba_pj_fwa_bqcz |
| 41 | $C_2^2:C_4$ | \( 1 + 12 T + 103 T^{2} + 894 T^{3} + 7405 T^{4} + 894 p T^{5} + 103 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.m_dz_bik_kyv |
| 43 | $C_2^2:C_4$ | \( 1 + 13 T + 51 T^{2} + 329 T^{3} + 3584 T^{4} + 329 p T^{5} + 51 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.n_bz_mr_fhw |
| 47 | $C_2^2:C_4$ | \( 1 + 4 T - 31 T^{2} + 118 T^{3} + 2789 T^{4} + 118 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.e_abf_eo_edh |
| 53 | $C_2^2:C_4$ | \( 1 + 10 T + 107 T^{2} + 930 T^{3} + 9569 T^{4} + 930 p T^{5} + 107 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.k_ed_bju_oeb |
| 59 | $C_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 492 T^{3} - 1595 T^{4} - 492 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.g_ax_asy_acjj |
| 61 | $D_{4}$ | \( ( 1 + 9 T + 141 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.s_nz_fjw_cdaz |
| 67 | $D_{4}$ | \( ( 1 - 21 T + 213 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.abq_bhj_arki_glvd |
| 71 | $C_2^2:C_4$ | \( 1 + 4 T + 25 T^{2} - 214 T^{3} + 309 T^{4} - 214 p T^{5} + 25 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.e_z_aig_lx |
| 73 | $C_2^2:C_4$ | \( 1 + 16 T + 113 T^{2} + 1340 T^{3} + 16621 T^{4} + 1340 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.q_ej_bzo_yph |
| 79 | $C_2^2:C_4$ | \( 1 + 25 T + 231 T^{2} + 1505 T^{3} + 12896 T^{4} + 1505 p T^{5} + 231 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.z_ix_cfx_tca |
| 83 | $C_2^2:C_4$ | \( 1 + 14 T + 13 T^{2} - 1190 T^{3} - 14559 T^{4} - 1190 p T^{5} + 13 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.o_n_abtu_avnz |
| 89 | $C_2^2:C_4$ | \( 1 + 20 T + 71 T^{2} - 690 T^{3} - 7699 T^{4} - 690 p T^{5} + 71 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.u_ct_abao_alkd |
| 97 | $C_2^2:C_4$ | \( 1 + 11 T + 129 T^{2} + 1717 T^{3} + 25484 T^{4} + 1717 p T^{5} + 129 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.l_ez_cob_blse |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.402827432779795554326115586582, −8.291582504217956885179014990884, −8.256092176426867550879165023526, −7.927212000598491814304272947466, −7.77533619376037357562298909311, −7.15647194716698543489626674394, −7.15614923358027101211237673668, −7.11251978654162560943512821716, −6.79979690146796384118079127719, −6.49995260150810746014593390640, −6.45621578421065071558332871180, −5.74779438113104558962505603528, −5.32282128051402100513393722980, −5.30894783746267694844056030388, −5.15025528250737749836886848841, −4.67797161212909359103004064568, −4.42181667811838097610147375996, −3.88595595822278332225027350274, −3.86799373952380710803283912787, −3.08328705809020121024596818416, −2.97707159919117071787994412453, −2.84874607240492592563692451075, −1.96084663612616139406909788397, −1.81520140033758280871464111810, −1.51130271985259989611170173492, 0, 0, 0, 0,
1.51130271985259989611170173492, 1.81520140033758280871464111810, 1.96084663612616139406909788397, 2.84874607240492592563692451075, 2.97707159919117071787994412453, 3.08328705809020121024596818416, 3.86799373952380710803283912787, 3.88595595822278332225027350274, 4.42181667811838097610147375996, 4.67797161212909359103004064568, 5.15025528250737749836886848841, 5.30894783746267694844056030388, 5.32282128051402100513393722980, 5.74779438113104558962505603528, 6.45621578421065071558332871180, 6.49995260150810746014593390640, 6.79979690146796384118079127719, 7.11251978654162560943512821716, 7.15614923358027101211237673668, 7.15647194716698543489626674394, 7.77533619376037357562298909311, 7.927212000598491814304272947466, 8.256092176426867550879165023526, 8.291582504217956885179014990884, 8.402827432779795554326115586582
