| L(s) = 1 | + 3·3-s − 3·5-s + 3·7-s + 8·9-s − 4·11-s + 5·13-s − 9·15-s − 9·17-s − 9·19-s + 9·21-s + 16·23-s + 10·25-s + 20·27-s + 17·29-s − 17·31-s − 12·33-s − 9·35-s − 11·37-s + 15·39-s + 11·41-s + 24·43-s − 24·45-s − 23·47-s + 12·49-s − 27·51-s − 3·53-s + 12·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 1.13·7-s + 8/3·9-s − 1.20·11-s + 1.38·13-s − 2.32·15-s − 2.18·17-s − 2.06·19-s + 1.96·21-s + 3.33·23-s + 2·25-s + 3.84·27-s + 3.15·29-s − 3.05·31-s − 2.08·33-s − 1.52·35-s − 1.80·37-s + 2.40·39-s + 1.71·41-s + 3.65·43-s − 3.57·45-s − 3.35·47-s + 12/7·49-s − 3.78·51-s − 0.412·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.102101257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.102101257\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2:C_4$ | \( 1 - p T + T^{2} + T^{3} + 4 T^{4} + p T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.3.ad_b_b_e |
| 5 | $C_2^2:C_4$ | \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 3 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.d_ab_ad_q |
| 7 | $C_2^2:C_4$ | \( 1 - 3 T - 3 T^{2} + 5 T^{3} + 36 T^{4} + 5 p T^{5} - 3 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ad_ad_f_bk |
| 13 | $C_2^2:C_4$ | \( 1 - 5 T + 27 T^{2} - 115 T^{3} + 584 T^{4} - 115 p T^{5} + 27 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.af_bb_ael_wm |
| 17 | $C_2^2:C_4$ | \( 1 + 9 T + 19 T^{2} + 3 T^{3} + 64 T^{4} + 3 p T^{5} + 19 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.j_t_d_cm |
| 19 | $C_2^2:C_4$ | \( 1 + 9 T + 17 T^{2} - 3 T^{3} + 100 T^{4} - 3 p T^{5} + 17 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.j_r_ad_dw |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.23.aq_hg_acai_lpu |
| 29 | $C_2^2:C_4$ | \( 1 - 17 T + 155 T^{2} - 1127 T^{3} + 6824 T^{4} - 1127 p T^{5} + 155 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ar_fz_abrj_kcm |
| 31 | $C_2^2:C_4$ | \( 1 + 17 T + 83 T^{2} - 191 T^{3} - 3320 T^{4} - 191 p T^{5} + 83 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.r_df_ahj_aexs |
| 37 | $C_2^2:C_4$ | \( 1 + 11 T + 39 T^{2} - 203 T^{3} - 3136 T^{4} - 203 p T^{5} + 39 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.l_bn_ahv_aeqq |
| 41 | $C_2^2:C_4$ | \( 1 - 11 T + 35 T^{2} + 271 T^{3} - 3856 T^{4} + 271 p T^{5} + 35 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.al_bj_kl_afsi |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ay_nk_afdw_bnes |
| 47 | $C_2^2:C_4$ | \( 1 + 23 T + 257 T^{2} + 2175 T^{3} + 16076 T^{4} + 2175 p T^{5} + 257 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.x_jx_dfr_xui |
| 53 | $C_2^2:C_4$ | \( 1 + 3 T - 49 T^{2} - 51 T^{3} + 2704 T^{4} - 51 p T^{5} - 49 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.d_abx_abz_eaa |
| 59 | $C_2^2:C_4$ | \( 1 - T - 43 T^{2} + 347 T^{3} + 2540 T^{4} + 347 p T^{5} - 43 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ab_abr_nj_dts |
| 61 | $C_2^2:C_4$ | \( 1 - 9 T - 25 T^{2} + 129 T^{3} + 2704 T^{4} + 129 p T^{5} - 25 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aj_az_ez_eaa |
| 67 | $D_{4}$ | \( ( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.bo_bfw_qqi_geig |
| 71 | $C_2^2:C_4$ | \( 1 - 13 T + 23 T^{2} - 301 T^{3} + 7280 T^{4} - 301 p T^{5} + 23 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.an_x_alp_kua |
| 73 | $C_2^2:C_4$ | \( 1 + T + 3 T^{2} + 515 T^{3} + 4976 T^{4} + 515 p T^{5} + 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.73.b_d_tv_hjk |
| 79 | $C_2^2:C_4$ | \( 1 - 7 T - 45 T^{2} - 167 T^{3} + 7784 T^{4} - 167 p T^{5} - 45 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ah_abt_agl_lnk |
| 83 | $C_2^2:C_4$ | \( 1 + 13 T + 31 T^{2} - 871 T^{3} - 12936 T^{4} - 871 p T^{5} + 31 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.n_bf_abhn_atdo |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_me_a_cijm |
| 97 | $C_2^2:C_4$ | \( 1 + 13 T + 107 T^{2} + 1775 T^{3} + 26416 T^{4} + 1775 p T^{5} + 107 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.n_ed_cqh_bnca |
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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
−7.57025374724597293794274668235, −7.30647818477623968821930764601, −6.95072781218081069338689337724, −6.92310272503728671225774660331, −6.91868292996879587219329009569, −6.44007393874956226474559205764, −6.38077005394140132515397574814, −5.79317453005183756910239791127, −5.58522336273474374778132283569, −5.14835602141372492087227309525, −4.93095025023622597527155674887, −4.63882364778984022897677815997, −4.57826988552891443907818082189, −4.22643028254228206914207850056, −4.19380161604062251216377041374, −3.84648569751503651028688426898, −3.46333186553752811707030483034, −3.00935954235365175978523557397, −2.94604717979169455616141608452, −2.56397547273055394927374749979, −2.50449470584052816248329512979, −1.72499198700020913123458509048, −1.68715400532632224429717378473, −1.12018302897269629398216967458, −0.57860853304774827232541025519,
0.57860853304774827232541025519, 1.12018302897269629398216967458, 1.68715400532632224429717378473, 1.72499198700020913123458509048, 2.50449470584052816248329512979, 2.56397547273055394927374749979, 2.94604717979169455616141608452, 3.00935954235365175978523557397, 3.46333186553752811707030483034, 3.84648569751503651028688426898, 4.19380161604062251216377041374, 4.22643028254228206914207850056, 4.57826988552891443907818082189, 4.63882364778984022897677815997, 4.93095025023622597527155674887, 5.14835602141372492087227309525, 5.58522336273474374778132283569, 5.79317453005183756910239791127, 6.38077005394140132515397574814, 6.44007393874956226474559205764, 6.91868292996879587219329009569, 6.92310272503728671225774660331, 6.95072781218081069338689337724, 7.30647818477623968821930764601, 7.57025374724597293794274668235
