| L(s) = 1 | + 2·2-s + 2·4-s − 4·5-s − 4·7-s + 2·8-s − 8·10-s − 8·13-s − 8·14-s + 16-s + 4·17-s − 4·19-s − 8·20-s + 8·23-s + 10·25-s − 16·26-s − 8·28-s − 4·29-s + 8·34-s + 16·35-s + 8·37-s − 8·38-s − 8·40-s − 4·41-s − 12·43-s + 16·46-s + 4·49-s + 20·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.78·5-s − 1.51·7-s + 0.707·8-s − 2.52·10-s − 2.21·13-s − 2.13·14-s + 1/4·16-s + 0.970·17-s − 0.917·19-s − 1.78·20-s + 1.66·23-s + 2·25-s − 3.13·26-s − 1.51·28-s − 0.742·29-s + 1.37·34-s + 2.70·35-s + 1.31·37-s − 1.29·38-s − 1.26·40-s − 0.624·41-s − 1.82·43-s + 2.35·46-s + 4/7·49-s + 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.979293314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.979293314\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - p T + p T^{2} - p T^{3} + 3 T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_c_ac_d |
| 7 | $C_2 \wr S_4$ | \( 1 + 4 T + 12 T^{2} + 20 T^{3} + 34 T^{4} + 20 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.e_m_u_bi |
| 13 | $C_2 \wr S_4$ | \( 1 + 8 T + 44 T^{2} + 144 T^{3} + 514 T^{4} + 144 p T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.i_bs_fo_tu |
| 17 | $C_2 \wr S_4$ | \( 1 - 4 T + 28 T^{2} + 36 T^{3} + 50 T^{4} + 36 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_bc_bk_by |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 20 T^{2} + 36 T^{3} + 326 T^{4} + 36 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.e_u_bk_mo |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 60 T^{2} - 296 T^{3} + 1510 T^{4} - 296 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ai_ci_alk_cgc |
| 29 | $C_2 \wr S_4$ | \( 1 + 4 T + 28 T^{2} + 108 T^{3} - 202 T^{4} + 108 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_bc_ee_ahu |
| 31 | $C_2 \wr S_4$ | \( 1 + 36 T^{2} + 192 T^{3} + 454 T^{4} + 192 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_bk_hk_rm |
| 37 | $C_2 \wr S_4$ | \( 1 - 8 T + 92 T^{2} - 664 T^{3} + 5046 T^{4} - 664 p T^{5} + 92 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_do_azo_hmc |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 44 T^{2} + 60 T^{3} + 2406 T^{4} + 60 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.e_bs_ci_doo |
| 43 | $C_2 \wr S_4$ | \( 1 + 12 T + 204 T^{2} + 36 p T^{3} + 13810 T^{4} + 36 p^{2} T^{5} + 204 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.m_hw_cho_ule |
| 47 | $C_2 \wr S_4$ | \( 1 + 60 T^{2} - 576 T^{3} + 646 T^{4} - 576 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_ci_awe_yw |
| 53 | $C_2 \wr S_4$ | \( 1 + 16 T + 252 T^{2} + 2416 T^{3} + 20854 T^{4} + 2416 p T^{5} + 252 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.q_js_doy_bewc |
| 59 | $C_2 \wr S_4$ | \( 1 - 24 T + 324 T^{2} - 2872 T^{3} + 386 p T^{4} - 2872 p T^{5} + 324 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ay_mm_aegm_bhry |
| 61 | $C_2 \wr S_4$ | \( 1 + 8 T + 140 T^{2} + 408 T^{3} + 7414 T^{4} + 408 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.i_fk_ps_kze |
| 67 | $C_2 \wr S_4$ | \( 1 + 44 T^{2} - 576 T^{3} + 3126 T^{4} - 576 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_bs_awe_eqg |
| 71 | $C_2 \wr S_4$ | \( 1 - 16 T + 324 T^{2} - 3280 T^{3} + 35686 T^{4} - 3280 p T^{5} + 324 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aq_mm_aewe_cauo |
| 73 | $C_2 \wr S_4$ | \( 1 + 8 T + 284 T^{2} + 1584 T^{3} + 418 p T^{4} + 1584 p T^{5} + 284 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.i_ky_ciy_btdq |
| 79 | $C_2 \wr S_4$ | \( 1 + 12 T + 308 T^{2} + 2652 T^{3} + 36342 T^{4} + 2652 p T^{5} + 308 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.m_lw_dya_cbtu |
| 83 | $C_2 \wr S_4$ | \( 1 - 8 T + 260 T^{2} - 1632 T^{3} + 31218 T^{4} - 1632 p T^{5} + 260 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ai_ka_acku_bues |
| 89 | $C_2 \wr S_4$ | \( 1 - 16 T + 332 T^{2} - 3760 T^{3} + 43974 T^{4} - 3760 p T^{5} + 332 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.aq_mu_afoq_cnbi |
| 97 | $C_2 \wr S_4$ | \( 1 - 8 T + 284 T^{2} - 1144 T^{3} + 33414 T^{4} - 1144 p T^{5} + 284 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ai_ky_absa_bxle |
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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
−5.47922591673372099419699497148, −5.45059878664150933828591519277, −5.37145921061925556692771621277, −5.21158547508398914567277208359, −4.90413705037577940465002747556, −4.65099194425477604175471530950, −4.57120408454952873098165801227, −4.56296200141175092144870365334, −4.15013021222600401388521250376, −3.92204831266492196312988662549, −3.81594575928137466314417948030, −3.78723290940588905866495478225, −3.21680006222864058862335229537, −3.18154660701695858170491332841, −3.10864741217229663290189037630, −2.83918891592811200790480591493, −2.80700350350083264859468399185, −2.35166843644661460367780381020, −2.15925475189847940063389887382, −1.88390725957979081882904169985, −1.48732058345190984566995134948, −1.32586236387447493654157080695, −0.63781020251627984940894761677, −0.49461552421871159581834058138, −0.30391324764261382475863252573,
0.30391324764261382475863252573, 0.49461552421871159581834058138, 0.63781020251627984940894761677, 1.32586236387447493654157080695, 1.48732058345190984566995134948, 1.88390725957979081882904169985, 2.15925475189847940063389887382, 2.35166843644661460367780381020, 2.80700350350083264859468399185, 2.83918891592811200790480591493, 3.10864741217229663290189037630, 3.18154660701695858170491332841, 3.21680006222864058862335229537, 3.78723290940588905866495478225, 3.81594575928137466314417948030, 3.92204831266492196312988662549, 4.15013021222600401388521250376, 4.56296200141175092144870365334, 4.57120408454952873098165801227, 4.65099194425477604175471530950, 4.90413705037577940465002747556, 5.21158547508398914567277208359, 5.37145921061925556692771621277, 5.45059878664150933828591519277, 5.47922591673372099419699497148
