| L(s) = 1 | + 6·3-s − 4·5-s + 15·9-s + 4·13-s − 24·15-s + 6·17-s + 6·23-s + 8·25-s + 18·27-s − 2·29-s + 4·37-s + 24·39-s + 12·41-s − 18·43-s − 60·45-s + 24·47-s − 9·49-s + 36·51-s + 8·53-s − 24·59-s + 14·61-s − 16·65-s + 24·67-s + 36·69-s − 4·73-s + 48·75-s + 9·81-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 1.78·5-s + 5·9-s + 1.10·13-s − 6.19·15-s + 1.45·17-s + 1.25·23-s + 8/5·25-s + 3.46·27-s − 0.371·29-s + 0.657·37-s + 3.84·39-s + 1.87·41-s − 2.74·43-s − 8.94·45-s + 3.50·47-s − 9/7·49-s + 5.04·51-s + 1.09·53-s − 3.12·59-s + 1.79·61-s − 1.98·65-s + 2.93·67-s + 4.33·69-s − 0.468·73-s + 5.54·75-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.600805792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.600805792\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T^{2} )^{2} \) | 4.3.ag_v_acc_ee |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.5.e_i_bc_dq |
| 7 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 24 T^{3} + 44 T^{4} - 24 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_j_ay_bs |
| 11 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 48 T^{3} + 20 T^{4} - 48 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_j_abw_u |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) | 4.17.ag_bh_aew_ns |
| 19 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 96 T^{3} - 124 T^{4} + 96 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_j_ds_aeu |
| 23 | $C_2^2$ | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ag_at_acc_cme |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 7 T^{2} - 94 T^{3} - 836 T^{4} - 94 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.c_ah_adq_abge |
| 31 | $C_2^3$ | \( 1 - 1858 T^{4} + p^{4} T^{8} \) | 4.31.a_a_a_actm |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 29 T^{2} + 288 T^{3} - 988 T^{4} + 288 p T^{5} + 29 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ae_bd_lc_abma |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 117 T^{2} - 840 T^{3} + 5804 T^{4} - 840 p T^{5} + 117 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.am_en_abgi_ipg |
| 43 | $C_2^2$ | \( ( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.s_gb_cec_rxs |
| 47 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2712 T^{3} + 21182 T^{4} - 2712 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ay_lc_aeai_bfis |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.ai_fk_abjk_qne |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 369 T^{2} + 3864 T^{3} + 33572 T^{4} + 3864 p T^{5} + 369 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.y_of_fsq_bxrg |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.ao_z_abak_ubs |
| 67 | $D_4\times C_2$ | \( 1 - 24 T + 153 T^{2} + 1224 T^{3} - 22972 T^{4} + 1224 p T^{5} + 153 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ay_fx_bvc_abhzo |
| 71 | $D_4\times C_2$ | \( 1 + 225 T^{2} + 120 T^{3} + 21884 T^{4} + 120 p T^{5} + 225 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ir_eq_bgjs |
| 73 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 204 T^{3} + 4718 T^{4} + 204 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.e_i_hw_gzm |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_alg_a_bxzy |
| 83 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2424 T^{3} + 20078 T^{4} + 2424 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.y_lc_dpg_bdsg |
| 89 | $D_4\times C_2$ | \( 1 - 20 T + 389 T^{2} - 4544 T^{3} + 52396 T^{4} - 4544 p T^{5} + 389 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.au_oz_agsu_czng |
| 97 | $D_4\times C_2$ | \( 1 - 28 T + 365 T^{2} - 2952 T^{3} + 23468 T^{4} - 2952 p T^{5} + 365 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.abc_ob_aejo_bisq |
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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
−7.63046801482156038816558298975, −7.17290023271907322804381174342, −6.97998171015726234435422869784, −6.96733886920786540244400590429, −6.54926599809224607144172061007, −6.39189906939493757440856948695, −5.81084344492153668192864222557, −5.64374639885029737853821012571, −5.54826190244780599556159093742, −5.20422658703256500110043203086, −4.82684301526019332470839086030, −4.41532582102828409390877485297, −4.28960833577309569336149140297, −3.93033827119975917231371832627, −3.90832588543808264551259276136, −3.45351122149051911642192100138, −3.28289887884746019437820782713, −3.20862384057814561820448374351, −2.88359547518147559494453296008, −2.67504039734212547714217195560, −2.42651308347083614319324776845, −1.89141168197249455123154718489, −1.64918140213356923316355568549, −0.937699133094478613661233250109, −0.71048750829350574556038074319,
0.71048750829350574556038074319, 0.937699133094478613661233250109, 1.64918140213356923316355568549, 1.89141168197249455123154718489, 2.42651308347083614319324776845, 2.67504039734212547714217195560, 2.88359547518147559494453296008, 3.20862384057814561820448374351, 3.28289887884746019437820782713, 3.45351122149051911642192100138, 3.90832588543808264551259276136, 3.93033827119975917231371832627, 4.28960833577309569336149140297, 4.41532582102828409390877485297, 4.82684301526019332470839086030, 5.20422658703256500110043203086, 5.54826190244780599556159093742, 5.64374639885029737853821012571, 5.81084344492153668192864222557, 6.39189906939493757440856948695, 6.54926599809224607144172061007, 6.96733886920786540244400590429, 6.97998171015726234435422869784, 7.17290023271907322804381174342, 7.63046801482156038816558298975
