| L(s) = 1 | + 7·5-s + 3·7-s − 11·11-s − 13-s + 6·17-s + 12·19-s − 8·23-s + 20·25-s + 10·29-s + 10·31-s + 21·35-s + 37-s + 15·41-s + 20·43-s + 5·47-s − 3·49-s − 26·53-s − 77·55-s − 2·59-s − 61-s − 7·65-s − 14·67-s − 5·71-s − 6·73-s − 33·77-s − 7·79-s + 83-s + ⋯ |
| L(s) = 1 | + 3.13·5-s + 1.13·7-s − 3.31·11-s − 0.277·13-s + 1.45·17-s + 2.75·19-s − 1.66·23-s + 4·25-s + 1.85·29-s + 1.79·31-s + 3.54·35-s + 0.164·37-s + 2.34·41-s + 3.04·43-s + 0.729·47-s − 3/7·49-s − 3.57·53-s − 10.3·55-s − 0.260·59-s − 0.128·61-s − 0.868·65-s − 1.71·67-s − 0.593·71-s − 0.702·73-s − 3.76·77-s − 0.787·79-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.892134863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.892134863\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_4$ | \( 1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2:C_4$ | \( 1 - 7 T + 29 T^{2} - 93 T^{3} + 236 T^{4} - 93 p T^{5} + 29 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ah_bd_adp_jc |
| 7 | $C_2^2:C_4$ | \( 1 - 3 T + 12 T^{2} - 5 p T^{3} + 141 T^{4} - 5 p^{2} T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ad_m_abj_fl |
| 13 | $C_2^2:C_4$ | \( 1 + T + 18 T^{2} + 5 T^{3} + 251 T^{4} + 5 p T^{5} + 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.13.b_s_f_jr |
| 17 | $C_2^2:C_4$ | \( 1 - 6 T - T^{2} + 108 T^{3} - 491 T^{4} + 108 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_ab_ee_asx |
| 19 | $C_2^2:C_4$ | \( 1 - 12 T + 75 T^{2} - 422 T^{3} + 2139 T^{4} - 422 p T^{5} + 75 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.am_cx_aqg_deh |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.i_ec_uy_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.ak_bf_ahs_csb |
| 31 | $C_2^2:C_4$ | \( 1 - 10 T + 29 T^{2} + 190 T^{3} - 2369 T^{4} + 190 p T^{5} + 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ak_bd_hi_adnd |
| 37 | $C_2^2:C_4$ | \( 1 - T + 14 T^{2} + 103 T^{3} + 739 T^{4} + 103 p T^{5} + 14 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ab_o_dz_bcl |
| 41 | $C_2^2:C_4$ | \( 1 - 15 T + 44 T^{2} + 435 T^{3} - 4649 T^{4} + 435 p T^{5} + 44 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ap_bs_qt_agwv |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 91 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.au_kw_adzc_beln |
| 47 | $C_2^2:C_4$ | \( 1 - 5 T - 37 T^{2} + 5 p T^{3} + 824 T^{4} + 5 p^{2} T^{5} - 37 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.af_abl_jb_bfs |
| 53 | $C_2^2:C_4$ | \( 1 + 26 T + 263 T^{2} + 1320 T^{3} + 5801 T^{4} + 1320 p T^{5} + 263 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ba_kd_byu_ipd |
| 59 | $C_2^2:C_4$ | \( 1 + 2 T + 5 T^{2} + 402 T^{3} + 4019 T^{4} + 402 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.c_f_pm_fyp |
| 61 | $C_2^2:C_4$ | \( 1 + T - 55 T^{2} + 179 T^{3} + 3844 T^{4} + 179 p T^{5} - 55 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.61.b_acd_gx_frw |
| 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 | $C_2^2:C_4$ | \( 1 + 5 T - 61 T^{2} - 5 p T^{3} + 2936 T^{4} - 5 p^{2} T^{5} - 61 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.f_acj_anr_eiy |
| 73 | $C_2^2:C_4$ | \( 1 + 6 T - 57 T^{2} - 130 T^{3} + 4761 T^{4} - 130 p T^{5} - 57 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.g_acf_afa_hbd |
| 79 | $C_2^2:C_4$ | \( 1 + 7 T + 80 T^{2} + 467 T^{3} + 1309 T^{4} + 467 p T^{5} + 80 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.h_dc_rz_byj |
| 83 | $C_2^2:C_4$ | \( 1 - T + 98 T^{2} - 295 T^{3} + 6881 T^{4} - 295 p T^{5} + 98 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ab_du_alj_ker |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.89.e_ny_bpg_ctxb |
| 97 | $C_2^2:C_4$ | \( 1 + 13 T - 3 T^{2} + 275 T^{3} + 12116 T^{4} + 275 p T^{5} - 3 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.n_ad_kp_rya |
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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
−8.080011065051070081025793445283, −7.80741524380577023545258765176, −7.75413424052055130150511028157, −7.70781291584179034552219271839, −7.36522311145941299570911195589, −7.01793187072569740014196318802, −6.38464440367668299828377706414, −6.33493516452842278553500034223, −5.89682151781803570345903390949, −5.83730220338691573113281123472, −5.71772404133654698623412320939, −5.36614886679301631462200702690, −5.28601675650963109582570121040, −4.93198934925697264532986774986, −4.69388884092043938640256424435, −4.29634760828092605867898064219, −4.19388352506057082099240223189, −3.10162850736509162765078815712, −3.04495654349904975591788200225, −2.79372108428957246267813875803, −2.57695755750749628541181171258, −2.24583532362425862651233854821, −1.70700392044004889295753684338, −1.38607121349315271436017041686, −0.892661504071150381320005476610,
0.892661504071150381320005476610, 1.38607121349315271436017041686, 1.70700392044004889295753684338, 2.24583532362425862651233854821, 2.57695755750749628541181171258, 2.79372108428957246267813875803, 3.04495654349904975591788200225, 3.10162850736509162765078815712, 4.19388352506057082099240223189, 4.29634760828092605867898064219, 4.69388884092043938640256424435, 4.93198934925697264532986774986, 5.28601675650963109582570121040, 5.36614886679301631462200702690, 5.71772404133654698623412320939, 5.83730220338691573113281123472, 5.89682151781803570345903390949, 6.33493516452842278553500034223, 6.38464440367668299828377706414, 7.01793187072569740014196318802, 7.36522311145941299570911195589, 7.70781291584179034552219271839, 7.75413424052055130150511028157, 7.80741524380577023545258765176, 8.080011065051070081025793445283
