| L(s) = 1 | + 3-s + 8·5-s + 7·7-s − 9·13-s + 8·15-s + 17-s + 13·19-s + 7·21-s + 8·23-s + 35·25-s − 10·29-s + 5·31-s + 56·35-s + 11·37-s − 9·39-s − 5·41-s − 20·43-s + 37·49-s + 51-s − 19·53-s + 13·57-s − 13·59-s − 4·61-s − 72·65-s − 14·67-s + 8·69-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.57·5-s + 2.64·7-s − 2.49·13-s + 2.06·15-s + 0.242·17-s + 2.98·19-s + 1.52·21-s + 1.66·23-s + 7·25-s − 1.85·29-s + 0.898·31-s + 9.46·35-s + 1.80·37-s − 1.44·39-s − 0.780·41-s − 3.04·43-s + 37/7·49-s + 0.140·51-s − 2.60·53-s + 1.72·57-s − 1.69·59-s − 0.512·61-s − 8.93·65-s − 1.71·67-s + 0.963·69-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.01327898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.01327898\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) | |
| 11 | | \( 1 \) | |
| good | 5 | $C_2^2:C_4$ | \( 1 - 8 T + 29 T^{2} - 72 T^{3} + 161 T^{4} - 72 p T^{5} + 29 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ai_bd_acu_gf |
| 7 | $C_2^2:C_4$ | \( 1 - p T + 12 T^{2} + 25 T^{3} - 139 T^{4} + 25 p T^{5} + 12 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.7.ah_m_z_afj |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 115 p T^{5} + 18 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.j_s_ael_abej |
| 17 | $C_2^2:C_4$ | \( 1 - T - T^{2} + 53 T^{3} + 104 T^{4} + 53 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ab_ab_cb_ea |
| 19 | $C_2^2:C_4$ | \( 1 - 13 T + 75 T^{2} - 353 T^{3} + 1664 T^{4} - 353 p T^{5} + 75 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.an_cx_anp_cma |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ai_ec_auy_fqx |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.k_bf_hs_csb |
| 31 | $C_2^2:C_4$ | \( 1 - 5 T + 29 T^{2} - 85 T^{3} - 44 T^{4} - 85 p T^{5} + 29 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.af_bd_adh_abs |
| 37 | $C_2^2:C_4$ | \( 1 - 11 T + 14 T^{2} + 413 T^{3} - 3701 T^{4} + 413 p T^{5} + 14 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.al_o_px_afmj |
| 41 | $C_2^2:C_4$ | \( 1 + 5 T + 44 T^{2} + 335 T^{3} + 3551 T^{4} + 335 p T^{5} + 44 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.f_bs_mx_fgp |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.u_kw_dzc_beln |
| 47 | $C_2^2:C_4$ | \( 1 - 37 T^{2} - 210 T^{3} + 1999 T^{4} - 210 p T^{5} - 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abl_aic_cyx |
| 53 | $C_2^2:C_4$ | \( 1 + 19 T + 263 T^{2} + 2685 T^{3} + 22496 T^{4} + 2685 p T^{5} + 263 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.t_kd_dzh_bhhg |
| 59 | $C_2^2:C_4$ | \( 1 + 13 T + 5 T^{2} - 687 T^{3} - 5716 T^{4} - 687 p T^{5} + 5 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.n_f_abal_ailw |
| 61 | $C_2^2:C_4$ | \( 1 + 4 T - 55 T^{2} - 184 T^{3} + 2929 T^{4} - 184 p T^{5} - 55 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.e_acd_ahc_eir |
| 67 | $D_{4}$ | \( ( 1 + 7 T + 135 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.o_mh_eeu_bxyv |
| 71 | $C_2^2:C_4$ | \( 1 - 61 T^{2} + 330 T^{3} + 4711 T^{4} + 330 p T^{5} - 61 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_acj_ms_gzf |
| 73 | $C_2^2:C_4$ | \( 1 + 4 T - 57 T^{2} + 170 T^{3} + 6221 T^{4} + 170 p T^{5} - 57 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.e_acf_go_jfh |
| 79 | $C_2^2:C_4$ | \( 1 - 17 T + 80 T^{2} + 923 T^{3} - 17651 T^{4} + 923 p T^{5} + 80 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ar_dc_bjn_abacx |
| 83 | $C_2^2:C_4$ | \( 1 - 21 T + 98 T^{2} + 1725 T^{3} - 29639 T^{4} + 1725 p T^{5} + 98 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.av_du_coj_abrvz |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.89.ae_ny_abpg_ctxb |
| 97 | $C_2^2:C_4$ | \( 1 - 12 T - 3 T^{2} - 500 T^{3} + 14541 T^{4} - 500 p T^{5} - 3 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.am_ad_atg_vnh |
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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
−6.80269120575895825346546879184, −6.44643350773625401568692874658, −6.43577639545123256152269579587, −5.94574767110797020533495865719, −5.82271407329673163728106334565, −5.73972268359706242541728341771, −5.44178085925122388693932892792, −5.09250821540177995950010502438, −5.03895493278672741563355270552, −4.85866180226035427214763168660, −4.79154597316741634561137024958, −4.70709157147341572135878207183, −4.39268154961595935675342281295, −3.65920475798492203702218745827, −3.40306754220291951483772258890, −3.13085039811564968384417726081, −3.03184623066341338528922963449, −2.67852532004189957296935906325, −2.34037600454252651588676793679, −2.16660782672772049825811370098, −1.92530672836687330750981561592, −1.62212526928790415490732355555, −1.44361712997005719225404500877, −1.17789653036762856254497972788, −0.69679904787164105524855456559,
0.69679904787164105524855456559, 1.17789653036762856254497972788, 1.44361712997005719225404500877, 1.62212526928790415490732355555, 1.92530672836687330750981561592, 2.16660782672772049825811370098, 2.34037600454252651588676793679, 2.67852532004189957296935906325, 3.03184623066341338528922963449, 3.13085039811564968384417726081, 3.40306754220291951483772258890, 3.65920475798492203702218745827, 4.39268154961595935675342281295, 4.70709157147341572135878207183, 4.79154597316741634561137024958, 4.85866180226035427214763168660, 5.03895493278672741563355270552, 5.09250821540177995950010502438, 5.44178085925122388693932892792, 5.73972268359706242541728341771, 5.82271407329673163728106334565, 5.94574767110797020533495865719, 6.43577639545123256152269579587, 6.44643350773625401568692874658, 6.80269120575895825346546879184
