| L(s) = 1 | − 2·2-s + 4-s + 5-s − 3·7-s + 2·8-s − 2·10-s − 8·11-s + 3·13-s + 6·14-s − 4·16-s − 9·17-s + 2·19-s + 20-s + 16·22-s + 2·25-s − 6·26-s − 3·28-s − 6·29-s − 26·31-s + 2·32-s + 18·34-s − 3·35-s + 7·37-s − 4·38-s + 2·40-s − 3·41-s − 26·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s + 0.707·8-s − 0.632·10-s − 2.41·11-s + 0.832·13-s + 1.60·14-s − 16-s − 2.18·17-s + 0.458·19-s + 0.223·20-s + 3.41·22-s + 2/5·25-s − 1.17·26-s − 0.566·28-s − 1.11·29-s − 4.66·31-s + 0.353·32-s + 3.08·34-s − 0.507·35-s + 1.15·37-s − 0.648·38-s + 0.316·40-s − 0.468·41-s − 3.96·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1782961527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1782961527\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ab_ab_i_aba |
| 7 | $D_4\times C_2$ | \( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 18 p T^{5} + p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.d_b_as_abw |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.11.i_cq_lk_bww |
| 13 | $D_4\times C_2$ | \( 1 - 3 T - 11 T^{2} + 18 T^{3} + 114 T^{4} + 18 p T^{5} - 11 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ad_al_s_ek |
| 17 | $D_4\times C_2$ | \( 1 + 9 T + 35 T^{2} + 108 T^{3} + 450 T^{4} + 108 p T^{5} + 35 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.j_bj_ee_ri |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 2 T^{2} + 64 T^{3} - 401 T^{4} + 64 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ac_ac_cm_apl |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.23.a_do_a_esc |
| 29 | $D_{4}$ | \( ( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.g_ej_ss_hgu |
| 31 | $D_{4}$ | \( ( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ba_nx_exa_bfzk |
| 41 | $D_4\times C_2$ | \( 1 + 3 T - 67 T^{2} - 18 T^{3} + 3726 T^{4} - 18 p T^{5} - 67 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.d_acp_as_fni |
| 43 | $D_{4}$ | \( ( 1 + 13 T + 120 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ba_pt_gha_bwhc |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.47.i_ie_bsq_wze |
| 53 | $C_2^3$ | \( 1 + 26 T^{2} - 2133 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ba_a_adeb |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.i_acs_ey_pvn |
| 61 | $D_4\times C_2$ | \( 1 - T - 47 T^{2} + 74 T^{3} - 1478 T^{4} + 74 p T^{5} - 47 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ab_abv_cw_acew |
| 67 | $D_4\times C_2$ | \( 1 - 9 T + T^{2} + 486 T^{3} - 3516 T^{4} + 486 p T^{5} + p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.aj_b_ss_affg |
| 71 | $C_2^3$ | \( 1 - 10 T^{2} - 4941 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ak_a_ahib |
| 73 | $D_{4}$ | \( ( 1 - 11 T + 168 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.aw_rp_ahvy_dfqy |
| 79 | $D_4\times C_2$ | \( 1 - 3 T - 143 T^{2} + 18 T^{3} + 16152 T^{4} + 18 p T^{5} - 143 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ad_afn_s_xxg |
| 83 | $C_2^2$ | \( ( 1 - 8 T - 19 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.aq_ba_abnk_bkql |
| 89 | $D_4\times C_2$ | \( 1 - 13 T - 43 T^{2} - 442 T^{3} + 21982 T^{4} - 442 p T^{5} - 43 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.an_abr_ara_bgnm |
| 97 | $C_2^2$ | \( ( 1 + 161 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_mk_a_coet |
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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
−7.71227432957783973405319129947, −7.45861779093245605199772711702, −7.18851040889890530756424438043, −6.85991744377134526530857925141, −6.61238046984711373162120326806, −6.50727654455501587169628626169, −6.40117917461754493619122542297, −5.99531037176670568002648754365, −5.52745289105988487352002997138, −5.35799679450631430870363791177, −5.19824344144370949070611278026, −5.00353744265072309315259240100, −4.95731935134134837153916173285, −4.25425271714136295971313812855, −3.99698603976483273157925167795, −3.82281379231754222082983253769, −3.33471950088550554121081529062, −3.14534325015164058069313209400, −3.09370813617015847667392442196, −2.29816079087062599639772980808, −2.07124771388573396557134066804, −1.84796125687520303578304446809, −1.71899984880105193808024637883, −0.59715013873653678507239109182, −0.24314051823975801318320563392,
0.24314051823975801318320563392, 0.59715013873653678507239109182, 1.71899984880105193808024637883, 1.84796125687520303578304446809, 2.07124771388573396557134066804, 2.29816079087062599639772980808, 3.09370813617015847667392442196, 3.14534325015164058069313209400, 3.33471950088550554121081529062, 3.82281379231754222082983253769, 3.99698603976483273157925167795, 4.25425271714136295971313812855, 4.95731935134134837153916173285, 5.00353744265072309315259240100, 5.19824344144370949070611278026, 5.35799679450631430870363791177, 5.52745289105988487352002997138, 5.99531037176670568002648754365, 6.40117917461754493619122542297, 6.50727654455501587169628626169, 6.61238046984711373162120326806, 6.85991744377134526530857925141, 7.18851040889890530756424438043, 7.45861779093245605199772711702, 7.71227432957783973405319129947
