| L(s) = 1 | − 4·3-s − 5·5-s + 4·7-s + 3·9-s + 13-s + 20·15-s − 17-s − 16·19-s − 16·21-s + 12·23-s + 15·25-s + 10·27-s − 9·29-s − 2·31-s − 20·35-s − 7·37-s − 4·39-s + 7·41-s + 24·43-s − 15·45-s + 16·47-s + 7·49-s + 4·51-s + 27·53-s + 64·57-s + 20·59-s − 26·61-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2.23·5-s + 1.51·7-s + 9-s + 0.277·13-s + 5.16·15-s − 0.242·17-s − 3.67·19-s − 3.49·21-s + 2.50·23-s + 3·25-s + 1.92·27-s − 1.67·29-s − 0.359·31-s − 3.38·35-s − 1.15·37-s − 0.640·39-s + 1.09·41-s + 3.65·43-s − 2.23·45-s + 2.33·47-s + 49-s + 0.560·51-s + 3.70·53-s + 8.47·57-s + 2.60·59-s − 3.32·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2693662753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2693662753\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 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 |
| 5 | $C_4\times C_2$ | \( 1 + p T + 2 p T^{2} + p^{2} T^{3} + 3 p^{2} T^{4} + p^{3} T^{5} + 2 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.5.f_k_z_cx |
| 7 | $C_2^2:C_4$ | \( 1 - 4 T + 9 T^{2} - 38 T^{3} + 149 T^{4} - 38 p T^{5} + 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ae_j_abm_ft |
| 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.ab_s_af_jr |
| 17 | $C_4\times C_2$ | \( 1 + T - 16 T^{2} - 33 T^{3} + 239 T^{4} - 33 p T^{5} - 16 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.17.b_aq_abh_jf |
| 19 | $C_4\times C_2$ | \( 1 + 16 T + 117 T^{2} + 578 T^{3} + 2525 T^{4} + 578 p T^{5} + 117 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.q_en_wg_dtd |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.am_fg_abhs_hso |
| 29 | $C_2^2:C_4$ | \( 1 + 9 T + 2 T^{2} - 243 T^{3} - 1445 T^{4} - 243 p T^{5} + 2 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.j_c_ajj_acdp |
| 31 | $C_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 94 T^{3} + 665 T^{4} + 94 p T^{5} + 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.c_bh_dq_zp |
| 37 | $C_2^2:C_4$ | \( 1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 145 p T^{5} - 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.h_as_afp_oh |
| 41 | $C_2^2:C_4$ | \( 1 - 7 T + 28 T^{2} - 389 T^{3} + 3975 T^{4} - 389 p T^{5} + 28 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ah_bc_aoz_fwx |
| 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 | $C_2^2:C_4$ | \( 1 - 16 T + 89 T^{2} - 522 T^{3} + 4709 T^{4} - 522 p T^{5} + 89 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.aq_dl_auc_gzd |
| 53 | $C_2^2:C_4$ | \( 1 - 27 T + 326 T^{2} - 2571 T^{3} + 18139 T^{4} - 2571 p T^{5} + 326 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.abb_mo_adux_bavr |
| 59 | $C_2^2:C_4$ | \( 1 - 20 T + 101 T^{2} + 20 p T^{3} - 18439 T^{4} + 20 p^{2} T^{5} + 101 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.au_dx_btk_abbhf |
| 61 | $C_2^2:C_4$ | \( 1 + 26 T + 215 T^{2} - 116 T^{3} - 10691 T^{4} - 116 p T^{5} + 215 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ba_ih_aem_apvf |
| 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 | $C_4\times C_2$ | \( 1 - 4 T - 55 T^{2} + 504 T^{3} + 1889 T^{4} + 504 p T^{5} - 55 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ae_acd_tk_cur |
| 73 | $C_2^2:C_4$ | \( 1 + 18 T + 51 T^{2} - 1256 T^{3} - 16971 T^{4} - 1256 p T^{5} + 51 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.s_bz_abwi_azct |
| 79 | $C_2^2:C_4$ | \( 1 - 10 T - 39 T^{2} + 10 p T^{3} - 2839 T^{4} + 10 p^{2} T^{5} - 39 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ak_abn_bek_aeff |
| 83 | $C_2^2:C_4$ | \( 1 + 26 T + 293 T^{2} + 2810 T^{3} + 28241 T^{4} + 2810 p T^{5} + 293 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ba_lh_eec_bpuf |
| 89 | $D_{4}$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ae_c_ano_ymt |
| 97 | $C_2^2:C_4$ | \( 1 + 23 T + 152 T^{2} + 625 T^{3} + 7951 T^{4} + 625 p T^{5} + 152 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.x_fw_yb_ltv |
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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
−7.18323943912959975366668063243, −7.09190624454322674520801151687, −6.91966957585438045431594506375, −6.24167042191870860365925790908, −6.19908158697424643421205661515, −5.99391923096880165217040326905, −5.84303951296651194035829911721, −5.52334424647686056193280315036, −5.50525669024145918136587881499, −5.02399253648605318678868001626, −4.91954889545029409932750636262, −4.64135579184794949776816798490, −4.32958950669168270750452223938, −4.22579951780246722986993915786, −3.94256364127787907736094132793, −3.75121427127297958853566188987, −3.68390527296291634452348740583, −2.80779875919634477379150544825, −2.58041096389708111426869710849, −2.56491013086633090863447980835, −2.18119591254881583157660184019, −1.34390447753281731827828751827, −1.23048101187121523318580826462, −0.53342010509712304411337473245, −0.28902981943076688392537877776,
0.28902981943076688392537877776, 0.53342010509712304411337473245, 1.23048101187121523318580826462, 1.34390447753281731827828751827, 2.18119591254881583157660184019, 2.56491013086633090863447980835, 2.58041096389708111426869710849, 2.80779875919634477379150544825, 3.68390527296291634452348740583, 3.75121427127297958853566188987, 3.94256364127787907736094132793, 4.22579951780246722986993915786, 4.32958950669168270750452223938, 4.64135579184794949776816798490, 4.91954889545029409932750636262, 5.02399253648605318678868001626, 5.50525669024145918136587881499, 5.52334424647686056193280315036, 5.84303951296651194035829911721, 5.99391923096880165217040326905, 6.19908158697424643421205661515, 6.24167042191870860365925790908, 6.91966957585438045431594506375, 7.09190624454322674520801151687, 7.18323943912959975366668063243
