| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 4·7-s − 2·8-s + 8·10-s + 4·11-s + 8·13-s − 8·14-s + 16-s − 4·17-s + 4·19-s − 8·20-s − 8·22-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 + 8·44-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 + 1.20·11-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 − 1.70·22-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 + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.490682470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490682470\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 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.c_c_c_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.ae_m_au_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.ai_bs_afo_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.e_bc_abk_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.ae_u_abk_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.ae_bc_aee_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.ae_bs_aci_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.am_hw_acho_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.ai_fk_aps_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.ai_ky_aciy_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.am_lw_adya_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.i_ka_cku_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
−8.041338260525170382927126132302, −7.70230530208883755234777368678, −7.62955082757483303126641867451, −7.34877680521384303159300218187, −6.86678679431246136779381990506, −6.79591229519875681372710899646, −6.77348057762279059228928202843, −6.25239210996437674214797573033, −6.02097899174936621404981621919, −5.91315024342879394175970033775, −5.31200258090562848770761218992, −5.01828025470229711006454113063, −4.96322108299691404875600581196, −4.46482881184923673252292304455, −4.30061259606992361945299167373, −4.03931269752574935162646005868, −3.68287402982119389317877758559, −3.36804369237588573730318615821, −3.35089061988114777181261457884, −2.57200487051432136500285185361, −2.39522899981451333800254076348, −1.86928691673103648163089535823, −1.11135507383157787584314474706, −1.05758285324307008203378166171, −0.77280509479617204392960142004,
0.77280509479617204392960142004, 1.05758285324307008203378166171, 1.11135507383157787584314474706, 1.86928691673103648163089535823, 2.39522899981451333800254076348, 2.57200487051432136500285185361, 3.35089061988114777181261457884, 3.36804369237588573730318615821, 3.68287402982119389317877758559, 4.03931269752574935162646005868, 4.30061259606992361945299167373, 4.46482881184923673252292304455, 4.96322108299691404875600581196, 5.01828025470229711006454113063, 5.31200258090562848770761218992, 5.91315024342879394175970033775, 6.02097899174936621404981621919, 6.25239210996437674214797573033, 6.77348057762279059228928202843, 6.79591229519875681372710899646, 6.86678679431246136779381990506, 7.34877680521384303159300218187, 7.62955082757483303126641867451, 7.70230530208883755234777368678, 8.041338260525170382927126132302
