| L(s) = 1 | − 2·2-s + 4-s − 5-s − 4·7-s + 2·8-s + 2·10-s + 18·11-s + 5·13-s + 8·14-s − 4·16-s + 4·17-s + 10·19-s − 20-s − 36·22-s + 3·23-s + 6·25-s − 10·26-s − 4·28-s − 4·29-s − 8·31-s + 2·32-s − 8·34-s + 4·35-s + 12·37-s − 20·38-s − 2·40-s − 3·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.707·8-s + 0.632·10-s + 5.42·11-s + 1.38·13-s + 2.13·14-s − 16-s + 0.970·17-s + 2.29·19-s − 0.223·20-s − 7.67·22-s + 0.625·23-s + 6/5·25-s − 1.96·26-s − 0.755·28-s − 0.742·29-s − 1.43·31-s + 0.353·32-s − 1.37·34-s + 0.676·35-s + 1.97·37-s − 3.24·38-s − 0.316·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.403909725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.403909725\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 + T - p T^{2} - 4 T^{3} + 6 T^{4} - 4 p T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.5.b_af_ae_g |
| 11 | $C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.as_gb_abhy_fdk |
| 13 | $D_4\times C_2$ | \( 1 - 5 T - 3 T^{2} - 10 T^{3} + 290 T^{4} - 10 p T^{5} - 3 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.af_ad_ak_le |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ae_aw_aq_bhz |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 35 T^{2} + 6 T^{3} + 1200 T^{4} + 6 p T^{5} - 35 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ad_abj_g_bue |
| 29 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 256 T^{3} - 1269 T^{4} - 256 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_w_ajw_abwv |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.31.i_fs_bdw_ktq |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.am_gm_abvo_oli |
| 41 | $D_4\times C_2$ | \( 1 + 3 T - 37 T^{2} - 108 T^{3} + 66 T^{4} - 108 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.d_abl_aee_co |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 42 T^{2} + 48 T^{3} + 2687 T^{4} + 48 p T^{5} - 42 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ag_abq_bw_dzj |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - 50 T^{2} - 48 T^{3} + 3495 T^{4} - 48 p T^{5} - 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.g_aby_abw_fel |
| 53 | $C_4\times C_2$ | \( 1 - 10 T - 14 T^{2} - 80 T^{3} + 4887 T^{4} - 80 p T^{5} - 14 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ak_ao_adc_hfz |
| 59 | $C_2^2$ | \( ( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.aw_jl_adyk_blcu |
| 61 | $D_4\times C_2$ | \( 1 - 25 T + 351 T^{2} - 3800 T^{3} + 33230 T^{4} - 3800 p T^{5} + 351 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.az_nn_afqe_bxec |
| 67 | $D_4\times C_2$ | \( 1 + 5 T - 77 T^{2} - 160 T^{3} + 4240 T^{4} - 160 p T^{5} - 77 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.f_acz_age_ghc |
| 71 | $D_4\times C_2$ | \( 1 + 9 T + 25 T^{2} - 774 T^{3} - 7656 T^{4} - 774 p T^{5} + 25 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.j_z_abdu_alim |
| 73 | $D_4\times C_2$ | \( 1 + 5 T - 123 T^{2} + 10 T^{3} + 14750 T^{4} + 10 p T^{5} - 123 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.f_aet_k_vvi |
| 79 | $D_4\times C_2$ | \( 1 + 12 T + 18 T^{2} - 384 T^{3} - 1741 T^{4} - 384 p T^{5} + 18 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.m_s_aou_acoz |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.83.ae_na_abmm_ckqd |
| 89 | $D_4\times C_2$ | \( 1 - 6 T + 2 T^{2} + 864 T^{3} - 9969 T^{4} + 864 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ag_c_bhg_aotl |
| 97 | $D_4\times C_2$ | \( 1 + 11 T - 99 T^{2} + 286 T^{3} + 27254 T^{4} + 286 p T^{5} - 99 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.l_adv_la_boig |
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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
−6.47963412920779208528427036771, −6.25715422080365604803487419331, −5.95729375774970234234268058011, −5.90127345890611194858371086667, −5.54741426879088016520798928923, −5.50528822272961489517414855183, −5.32988650226675253521004849195, −4.82640273685263657443447725666, −4.41826072827652184525098123939, −4.38099192564881024149751080732, −4.17727393694771425991049708255, −3.79354878263055185436039834404, −3.75627806766967221011506321147, −3.61422334077751915354327139361, −3.49295072187826566981952417059, −3.18970534056385417855520848493, −2.91902437203170988679712587121, −2.62859123685951182075750282944, −2.03467201470873476474170204458, −1.82891959303098958997562013213, −1.45361010198442375818114668864, −1.11833894570779056478690691783, −0.972123781395676078099789040018, −0.922990930386331726563294859439, −0.58832842907254813944153568622,
0.58832842907254813944153568622, 0.922990930386331726563294859439, 0.972123781395676078099789040018, 1.11833894570779056478690691783, 1.45361010198442375818114668864, 1.82891959303098958997562013213, 2.03467201470873476474170204458, 2.62859123685951182075750282944, 2.91902437203170988679712587121, 3.18970534056385417855520848493, 3.49295072187826566981952417059, 3.61422334077751915354327139361, 3.75627806766967221011506321147, 3.79354878263055185436039834404, 4.17727393694771425991049708255, 4.38099192564881024149751080732, 4.41826072827652184525098123939, 4.82640273685263657443447725666, 5.32988650226675253521004849195, 5.50528822272961489517414855183, 5.54741426879088016520798928923, 5.90127345890611194858371086667, 5.95729375774970234234268058011, 6.25715422080365604803487419331, 6.47963412920779208528427036771
