| L(s) = 1 | − 4·3-s + 5-s − 7-s + 3·9-s − 11-s + 3·13-s − 4·15-s + 8·17-s + 14·19-s + 4·21-s − 14·23-s + 10·27-s − 6·29-s − 10·31-s + 4·33-s − 35-s − 10·37-s − 12·39-s − 9·41-s + 28·43-s + 3·45-s − 14·47-s + 2·49-s − 32·51-s + 6·53-s − 55-s − 56·57-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.447·5-s − 0.377·7-s + 9-s − 0.301·11-s + 0.832·13-s − 1.03·15-s + 1.94·17-s + 3.21·19-s + 0.872·21-s − 2.91·23-s + 1.92·27-s − 1.11·29-s − 1.79·31-s + 0.696·33-s − 0.169·35-s − 1.64·37-s − 1.92·39-s − 1.40·41-s + 4.26·43-s + 0.447·45-s − 2.04·47-s + 2/7·49-s − 4.48·51-s + 0.824·53-s − 0.134·55-s − 7.41·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4713701295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4713701295\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) | |
| 11 | $C_4$ | \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 10 p T^{3} + 61 T^{4} + 10 p^{2} T^{5} + 13 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.e_n_be_cj |
| 7 | $C_2^2:C_4$ | \( 1 + T - T^{2} + 17 T^{3} + 64 T^{4} + 17 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.7.b_ab_r_cm |
| 13 | $C_2^2:C_4$ | \( 1 - 3 T - 9 T^{2} + 11 T^{3} + 144 T^{4} + 11 p T^{5} - 9 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ad_aj_l_fo |
| 17 | $C_2^2:C_4$ | \( 1 - 8 T + 7 T^{2} + 110 T^{3} - 579 T^{4} + 110 p T^{5} + 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ai_h_eg_awh |
| 19 | $C_2^2:C_4$ | \( 1 - 14 T + 77 T^{2} - 262 T^{3} + 955 T^{4} - 262 p T^{5} + 77 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ao_cz_akc_bkt |
| 23 | $D_{4}$ | \( ( 1 + 7 T + 47 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.o_fn_bls_iej |
| 29 | $C_2^2:C_4$ | \( 1 + 6 T - 13 T^{2} - 42 T^{3} + 625 T^{4} - 42 p T^{5} - 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.g_an_abq_yb |
| 31 | $C_2^2:C_4$ | \( 1 + 10 T + 9 T^{2} - 10 p T^{3} - 2359 T^{4} - 10 p^{2} T^{5} + 9 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.k_j_aly_admt |
| 37 | $C_2^2:C_4$ | \( 1 + 10 T + 3 T^{2} - 160 T^{3} - 271 T^{4} - 160 p T^{5} + 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.k_d_age_akl |
| 41 | $C_2^2:C_4$ | \( 1 + 9 T - 5 T^{2} - 69 T^{3} + 1024 T^{4} - 69 p T^{5} - 5 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.j_af_acr_bnk |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.abc_ro_ahei_cdko |
| 47 | $C_2^2:C_4$ | \( 1 + 14 T + 149 T^{2} + 1358 T^{3} + 11519 T^{4} + 1358 p T^{5} + 149 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.o_ft_cag_rbb |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T + 83 T^{2} - 480 T^{3} + 6481 T^{4} - 480 p T^{5} + 83 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ag_df_asm_jph |
| 59 | $C_2^2:C_4$ | \( 1 - 6 T - 43 T^{2} + 402 T^{3} + 475 T^{4} + 402 p T^{5} - 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ag_abr_pm_sh |
| 61 | $C_4\times C_2$ | \( 1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 1600 p T^{5} + 179 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.u_gx_cjo_wez |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.m_ma_dqq_bwpi |
| 71 | $C_2^2:C_4$ | \( 1 - 16 T + 65 T^{2} + 866 T^{3} - 14731 T^{4} + 866 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aq_cn_bhi_avup |
| 73 | $C_2^2:C_4$ | \( 1 + 20 T + 87 T^{2} - 530 T^{3} - 7411 T^{4} - 530 p T^{5} + 87 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.u_dj_auk_akzb |
| 79 | $C_2^2:C_4$ | \( 1 - 4 T + 17 T^{2} - 692 T^{3} + 9025 T^{4} - 692 p T^{5} + 17 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ae_r_abaq_njd |
| 83 | $C_2^2:C_4$ | \( 1 - 24 T + 133 T^{2} + 1710 T^{3} - 30659 T^{4} + 1710 p T^{5} + 133 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ay_fd_cnu_abtjf |
| 89 | $D_{4}$ | \( ( 1 - 5 T - 27 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ak_abd_axw_bfcn |
| 97 | $C_4\times C_2$ | \( 1 - 12 T + 47 T^{2} + 600 T^{3} - 11759 T^{4} + 600 p T^{5} + 47 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.am_bv_xc_arkh |
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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
−7.42881956736820521785702832750, −7.06233779471445237453648151727, −6.64381204638934028287994536325, −6.38402968599065687130744263661, −6.17219035149514119021436808836, −5.99575938949660192886926574651, −5.88874120244476650114791589831, −5.65694633471211819081122103975, −5.39713861190806417640492861228, −5.38476876915232035944229315899, −5.30946780195671178311501907032, −4.80063822540358980346091353767, −4.71616291776267492174288850347, −4.05696067116819184744226271208, −3.92269577987806202708512371254, −3.51607146808153072974886664655, −3.45877867266554865598593383211, −3.13838019675347849776470100700, −2.98506324060997226022653458069, −2.20271126909133908444557902018, −2.16799708877162877619198917754, −1.52077292213956234212611354254, −1.33337557006603995058735957164, −0.72729118320162905364509830014, −0.28246081713366663937327098389,
0.28246081713366663937327098389, 0.72729118320162905364509830014, 1.33337557006603995058735957164, 1.52077292213956234212611354254, 2.16799708877162877619198917754, 2.20271126909133908444557902018, 2.98506324060997226022653458069, 3.13838019675347849776470100700, 3.45877867266554865598593383211, 3.51607146808153072974886664655, 3.92269577987806202708512371254, 4.05696067116819184744226271208, 4.71616291776267492174288850347, 4.80063822540358980346091353767, 5.30946780195671178311501907032, 5.38476876915232035944229315899, 5.39713861190806417640492861228, 5.65694633471211819081122103975, 5.88874120244476650114791589831, 5.99575938949660192886926574651, 6.17219035149514119021436808836, 6.38402968599065687130744263661, 6.64381204638934028287994536325, 7.06233779471445237453648151727, 7.42881956736820521785702832750
