| L(s) = 1 | + 3-s + 5-s − 2·7-s + 3·13-s + 15-s + 7·17-s − 4·19-s − 2·21-s − 8·23-s + 5·25-s − 3·29-s − 2·35-s − 9·37-s + 3·39-s − 9·41-s − 8·43-s + 12·47-s + 7·49-s + 7·51-s + 13·53-s − 4·57-s + 6·59-s + 2·61-s + 3·65-s − 56·67-s − 8·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 0.832·13-s + 0.258·15-s + 1.69·17-s − 0.917·19-s − 0.436·21-s − 1.66·23-s + 25-s − 0.557·29-s − 0.338·35-s − 1.47·37-s + 0.480·39-s − 1.40·41-s − 1.21·43-s + 1.75·47-s + 49-s + 0.980·51-s + 1.78·53-s − 0.529·57-s + 0.781·59-s + 0.256·61-s + 0.372·65-s − 6.84·67-s − 0.963·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171222878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171222878\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) | |
| 11 | | \( 1 \) | |
| good | 5 | $C_4\times C_2$ | \( 1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} + 9 p T^{5} - 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ab_ae_j_l |
| 7 | $C_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 20 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.c_ad_au_at |
| 13 | $C_4\times C_2$ | \( 1 - 3 T - 4 T^{2} + 51 T^{3} - 101 T^{4} + 51 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ad_ae_bz_adx |
| 17 | $C_4\times C_2$ | \( 1 - 7 T + 32 T^{2} - 105 T^{3} + 191 T^{4} - 105 p T^{5} + 32 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ah_bg_aeb_hj |
| 19 | $C_4\times C_2$ | \( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 88 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.e_ad_adk_alj |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.23.i_em_wm_gje |
| 29 | $C_4\times C_2$ | \( 1 + 3 T - 20 T^{2} - 147 T^{3} + 139 T^{4} - 147 p T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.d_au_afr_fj |
| 31 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.31.a_abf_a_bkz |
| 37 | $C_4\times C_2$ | \( 1 + 9 T + 44 T^{2} + 63 T^{3} - 1061 T^{4} + 63 p T^{5} + 44 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.j_bs_cl_abov |
| 41 | $C_4\times C_2$ | \( 1 + 9 T + 40 T^{2} - 9 T^{3} - 1721 T^{4} - 9 p T^{5} + 40 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.j_bo_aj_acof |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.43.i_ho_boy_tms |
| 47 | $C_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 600 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.am_dt_axc_dxp |
| 53 | $C_4\times C_2$ | \( 1 - 13 T + 116 T^{2} - 819 T^{3} + 4499 T^{4} - 819 p T^{5} + 116 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.an_em_abfn_grb |
| 59 | $C_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 492 T^{3} - 1595 T^{4} + 492 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ag_ax_sy_acjj |
| 61 | $C_4\times C_2$ | \( 1 - 2 T - 57 T^{2} + 236 T^{3} + 3005 T^{4} + 236 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ac_acf_jc_elp |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) | 4.67.ce_cdo_bgxc_mruk |
| 71 | $C_4\times C_2$ | \( 1 - 6 T - 35 T^{2} + 636 T^{3} - 1331 T^{4} + 636 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ag_abj_ym_abzf |
| 73 | $C_4\times C_2$ | \( 1 + 6 T - 37 T^{2} - 660 T^{3} - 1259 T^{4} - 660 p T^{5} - 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.g_abl_azk_abwl |
| 79 | $C_4\times C_2$ | \( 1 - 16 T + 177 T^{2} - 1568 T^{3} + 11105 T^{4} - 1568 p T^{5} + 177 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.aq_gv_acii_qld |
| 83 | $C_4\times C_2$ | \( 1 - 8 T - 19 T^{2} + 816 T^{3} - 4951 T^{4} + 816 p T^{5} - 19 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ai_at_bfk_ahil |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.89.ae_ny_abpg_ctxb |
| 97 | $C_4\times C_2$ | \( 1 + 7 T - 48 T^{2} - 1015 T^{3} - 2449 T^{4} - 1015 p T^{5} - 48 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.h_abw_abnb_adqf |
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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
−6.95475502358097049688987112518, −6.63202460669086562962209054407, −6.15260379024122235375596035819, −6.00967830593368681444344091406, −5.99022851936361109107686147858, −5.84336668159497617154560614229, −5.60430383962789097437342101748, −5.40535806028369771109521253923, −4.80695464433314067391467184214, −4.78627192991650887754016231898, −4.78351326521346523724910428076, −4.18382437864247706863481501891, −4.02695638496569428725335211035, −3.71708477891755821522566568907, −3.59343128621729722763530880683, −3.36554830415191208917728593451, −3.00775365446401368363332543647, −2.93762309949590465904108138318, −2.60903694060433089684924866266, −2.00315215449568553826876664006, −1.91509518959696646201083725117, −1.84612753659751339482699235774, −1.19747968113164300815364370227, −0.918001740735555867566255356576, −0.28232861169980754270098328130,
0.28232861169980754270098328130, 0.918001740735555867566255356576, 1.19747968113164300815364370227, 1.84612753659751339482699235774, 1.91509518959696646201083725117, 2.00315215449568553826876664006, 2.60903694060433089684924866266, 2.93762309949590465904108138318, 3.00775365446401368363332543647, 3.36554830415191208917728593451, 3.59343128621729722763530880683, 3.71708477891755821522566568907, 4.02695638496569428725335211035, 4.18382437864247706863481501891, 4.78351326521346523724910428076, 4.78627192991650887754016231898, 4.80695464433314067391467184214, 5.40535806028369771109521253923, 5.60430383962789097437342101748, 5.84336668159497617154560614229, 5.99022851936361109107686147858, 6.00967830593368681444344091406, 6.15260379024122235375596035819, 6.63202460669086562962209054407, 6.95475502358097049688987112518
