| L(s) = 1 | + 2-s + 3·4-s − 3·5-s + 3·7-s + 4·8-s − 3·10-s + 5·11-s + 2·13-s + 3·14-s + 6·16-s − 8·17-s − 12·19-s − 9·20-s + 5·22-s − 5·23-s + 11·25-s + 2·26-s + 9·28-s − 5·29-s − 31-s + 6·32-s − 8·34-s − 9·35-s − 2·37-s − 12·38-s − 12·40-s − 11·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3/2·4-s − 1.34·5-s + 1.13·7-s + 1.41·8-s − 0.948·10-s + 1.50·11-s + 0.554·13-s + 0.801·14-s + 3/2·16-s − 1.94·17-s − 2.75·19-s − 2.01·20-s + 1.06·22-s − 1.04·23-s + 11/5·25-s + 0.392·26-s + 1.70·28-s − 0.928·29-s − 0.179·31-s + 1.06·32-s − 1.37·34-s − 1.52·35-s − 0.328·37-s − 1.94·38-s − 1.89·40-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.265713536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.265713536\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - T - p T^{2} + T^{3} + 3 T^{4} + p T^{5} - p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.2.ab_ac_b_d |
| 5 | $D_4\times C_2$ | \( 1 + 3 T - 2 T^{2} + 3 T^{3} + 51 T^{4} + 3 p T^{5} - 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.d_ac_d_bz |
| 7 | $D_4\times C_2$ | \( 1 - 3 T - 6 T^{2} - 3 T^{3} + 113 T^{4} - 3 p T^{5} - 6 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ad_ag_ad_ej |
| 11 | $D_4\times C_2$ | \( 1 - 5 T + 8 T^{2} + 25 T^{3} - 107 T^{4} + 25 p T^{5} + 8 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.af_i_z_aed |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.i_de_pk_dhb |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.m_dm_vg_eel |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - 16 T^{2} - 25 T^{3} + 577 T^{4} - 25 p T^{5} - 16 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.f_aq_az_wf |
| 29 | $D_4\times C_2$ | \( 1 + 5 T - 28 T^{2} - 25 T^{3} + 1243 T^{4} - 25 p T^{5} - 28 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.f_abc_az_bvv |
| 31 | $D_4\times C_2$ | \( 1 + T - 61 T^{3} - 991 T^{4} - 61 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) | 4.31.b_a_acj_abmd |
| 37 | $D_{4}$ | \( ( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.c_dj_ge_gxh |
| 41 | $C_4\times C_2$ | \( 1 + 11 T + 10 T^{2} + 319 T^{3} + 5679 T^{4} + 319 p T^{5} + 10 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.l_k_mh_ikl |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 111 T^{2} - 944 T^{3} + 8168 T^{4} - 944 p T^{5} + 111 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.aq_eh_abki_mce |
| 47 | $D_4\times C_2$ | \( 1 - 6 T - p T^{2} + 66 T^{3} + 2988 T^{4} + 66 p T^{5} - p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ag_abv_co_eky |
| 53 | $D_{4}$ | \( ( 1 + 14 T + 110 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.bc_qa_gto_ceyo |
| 59 | $D_4\times C_2$ | \( 1 + 2 T - 95 T^{2} - 38 T^{3} + 6084 T^{4} - 38 p T^{5} - 95 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.c_adr_abm_jaa |
| 61 | $D_4\times C_2$ | \( 1 - 5 T - 72 T^{2} + 125 T^{3} + 4163 T^{4} + 125 p T^{5} - 72 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.af_acu_ev_ged |
| 67 | $D_4\times C_2$ | \( 1 - 7 T - 66 T^{2} + 133 T^{3} + 5453 T^{4} + 133 p T^{5} - 66 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ah_aco_fd_ibt |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.71.u_qs_hbc_czep |
| 73 | $C_2^2$ | \( ( 1 + 21 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_bq_a_qkx |
| 79 | $C_2^3$ | \( 1 - 153 T^{2} + 17168 T^{4} - 153 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_afx_a_zki |
| 83 | $D_4\times C_2$ | \( 1 - 8 T + 7 T^{2} + 872 T^{3} - 8952 T^{4} + 872 p T^{5} + 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ai_h_bho_angi |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 179 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ag_od_acjw_cvfh |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 162 T^{2} + 64 T^{3} + 20723 T^{4} + 64 p T^{5} - 162 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ae_agg_cm_berb |
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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
−7.17052004329393402439930321952, −6.73795044461022738825259231447, −6.59792783299622358864146629239, −6.45148732625760120493749774548, −6.23617849419232246564260000730, −6.22209724642960178732059413467, −5.81501170156196494565746502534, −5.51036185766610577620848536359, −5.22777198392522840978067947749, −4.80096718324148790569826749682, −4.78948884075641872230065742543, −4.40675798462260491680681735737, −4.27042658331988496363768806275, −4.10948527386438783930002288240, −3.87968675319543277590500981507, −3.68868064114302962507631551500, −3.35615995503582759917659551635, −3.00069105260965291004724742362, −2.58804553422546964139525309135, −2.26814983336251940666991998974, −2.09474814469460978821497183148, −1.73573088395969596915403030918, −1.58689609929421983199651892011, −1.05546753601759007571873825017, −0.32810291929510281954554120322,
0.32810291929510281954554120322, 1.05546753601759007571873825017, 1.58689609929421983199651892011, 1.73573088395969596915403030918, 2.09474814469460978821497183148, 2.26814983336251940666991998974, 2.58804553422546964139525309135, 3.00069105260965291004724742362, 3.35615995503582759917659551635, 3.68868064114302962507631551500, 3.87968675319543277590500981507, 4.10948527386438783930002288240, 4.27042658331988496363768806275, 4.40675798462260491680681735737, 4.78948884075641872230065742543, 4.80096718324148790569826749682, 5.22777198392522840978067947749, 5.51036185766610577620848536359, 5.81501170156196494565746502534, 6.22209724642960178732059413467, 6.23617849419232246564260000730, 6.45148732625760120493749774548, 6.59792783299622358864146629239, 6.73795044461022738825259231447, 7.17052004329393402439930321952
