| L(s) = 1 | + 7-s − 5·9-s + 11-s − 13-s + 8·17-s + 7·19-s − 4·23-s + 27-s − 5·29-s + 14·31-s + 4·37-s + 3·41-s − 7·43-s + 12·47-s − 13·49-s − 10·53-s + 11·59-s + 22·61-s − 5·63-s + 12·67-s + 18·71-s − 9·73-s + 77-s + 25·79-s + 11·81-s − 7·83-s − 91-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 5/3·9-s + 0.301·11-s − 0.277·13-s + 1.94·17-s + 1.60·19-s − 0.834·23-s + 0.192·27-s − 0.928·29-s + 2.51·31-s + 0.657·37-s + 0.468·41-s − 1.06·43-s + 1.75·47-s − 1.85·49-s − 1.37·53-s + 1.43·59-s + 2.81·61-s − 0.629·63-s + 1.46·67-s + 2.13·71-s − 1.05·73-s + 0.113·77-s + 2.81·79-s + 11/9·81-s − 0.768·83-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.559853897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.559853897\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 5 T^{2} - T^{3} + 14 T^{4} - p T^{5} + 5 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_f_ab_o |
| 7 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} - 3 p T^{3} + 106 T^{4} - 3 p^{2} T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ab_o_av_ec |
| 11 | $C_2 \wr S_4$ | \( 1 - T + 18 T^{2} - 9 T^{3} + 274 T^{4} - 9 p T^{5} + 18 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ab_s_aj_ko |
| 13 | $C_2 \wr S_4$ | \( 1 + T + 21 T^{2} + 7 p T^{3} + 191 T^{4} + 7 p^{2} T^{5} + 21 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.13.b_v_dn_hj |
| 17 | $C_2 \wr S_4$ | \( 1 - 8 T + 64 T^{2} - 336 T^{3} + 1550 T^{4} - 336 p T^{5} + 64 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ai_cm_amy_chq |
| 19 | $C_2 \wr S_4$ | \( 1 - 7 T + 68 T^{2} - 339 T^{3} + 1934 T^{4} - 339 p T^{5} + 68 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ah_cq_anb_cwk |
| 29 | $C_2 \wr S_4$ | \( 1 + 5 T + 85 T^{2} + 303 T^{3} + 3155 T^{4} + 303 p T^{5} + 85 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.f_dh_lr_erj |
| 31 | $C_2 \wr S_4$ | \( 1 - 14 T + 173 T^{2} - 1329 T^{3} + 8794 T^{4} - 1329 p T^{5} + 173 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ao_gr_abzd_nag |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 56 T^{2} - 36 T^{3} + 1822 T^{4} - 36 p T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ae_ce_abk_csc |
| 41 | $C_2 \wr S_4$ | \( 1 - 3 T + 57 T^{2} - 195 T^{3} + 1693 T^{4} - 195 p T^{5} + 57 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ad_cf_ahn_cnd |
| 43 | $C_2 \wr S_4$ | \( 1 + 7 T + 164 T^{2} + 843 T^{3} + 10478 T^{4} + 843 p T^{5} + 164 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.h_gi_bgl_pna |
| 47 | $C_2 \wr S_4$ | \( 1 - 12 T + 155 T^{2} - 1089 T^{3} + 8910 T^{4} - 1089 p T^{5} + 155 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.am_fz_abpx_nes |
| 53 | $C_2 \wr S_4$ | \( 1 + 10 T + 196 T^{2} + 1422 T^{3} + 15446 T^{4} + 1422 p T^{5} + 196 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.k_ho_ccs_wwc |
| 59 | $C_2 \wr S_4$ | \( 1 - 11 T + 255 T^{2} - 1896 T^{3} + 23140 T^{4} - 1896 p T^{5} + 255 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.al_jv_acuy_biga |
| 61 | $C_2 \wr S_4$ | \( 1 - 22 T + 372 T^{2} - 4066 T^{3} + 37526 T^{4} - 4066 p T^{5} + 372 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aw_oi_agak_cdni |
| 67 | $C_2 \wr S_4$ | \( 1 - 12 T + 240 T^{2} - 2036 T^{3} + 24174 T^{4} - 2036 p T^{5} + 240 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.am_jg_adai_bjtu |
| 71 | $C_2 \wr S_4$ | \( 1 - 18 T + 351 T^{2} - 3507 T^{3} + 37960 T^{4} - 3507 p T^{5} + 351 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.as_nn_afex_ceea |
| 73 | $C_2 \wr S_4$ | \( 1 + 9 T + 273 T^{2} + 1813 T^{3} + 29061 T^{4} + 1813 p T^{5} + 273 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.j_kn_crt_bqzt |
| 79 | $C_2 \wr S_4$ | \( 1 - 25 T + 492 T^{2} - 6169 T^{3} + 65294 T^{4} - 6169 p T^{5} + 492 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.az_sy_ajdh_dspi |
| 83 | $C_2 \wr S_4$ | \( 1 + 7 T + 256 T^{2} + 1539 T^{3} + 28790 T^{4} + 1539 p T^{5} + 256 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.h_jw_chf_bqpi |
| 89 | $C_2 \wr S_4$ | \( 1 + 120 T^{2} + 1848 T^{3} + 1678 T^{4} + 1848 p T^{5} + 120 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_eq_ctc_cmo |
| 97 | $C_2 \wr S_4$ | \( 1 - 8 T + 168 T^{2} - 272 T^{3} + 12734 T^{4} - 272 p T^{5} + 168 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ai_gm_akm_svu |
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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
−6.42027350833503070371002059747, −6.13795723302372411961517253201, −5.86558213706513342482379828260, −5.72722961441655921460060869196, −5.66017587502654708850454568150, −5.27889610191501717222774177209, −5.12596993523503661204242475130, −5.06376061040392498474141557630, −4.96451518138018911294576151642, −4.34493769862659326794715372607, −4.23749972633114267541345518258, −4.19276969003558369395713495350, −3.67832707496253281941476014171, −3.43069748023418714399423698800, −3.31282129278808505274108566518, −3.19439333852046680510275179824, −2.96497661211467502769443272755, −2.50816468147174698192902501090, −2.33469314281314025074146866530, −2.10260580101694375109831768266, −1.88611302261101769169952079942, −1.22366771950101003094785309182, −1.08386624362675034798785605401, −0.67541949872324653309119287598, −0.52380850527112514778561993509,
0.52380850527112514778561993509, 0.67541949872324653309119287598, 1.08386624362675034798785605401, 1.22366771950101003094785309182, 1.88611302261101769169952079942, 2.10260580101694375109831768266, 2.33469314281314025074146866530, 2.50816468147174698192902501090, 2.96497661211467502769443272755, 3.19439333852046680510275179824, 3.31282129278808505274108566518, 3.43069748023418714399423698800, 3.67832707496253281941476014171, 4.19276969003558369395713495350, 4.23749972633114267541345518258, 4.34493769862659326794715372607, 4.96451518138018911294576151642, 5.06376061040392498474141557630, 5.12596993523503661204242475130, 5.27889610191501717222774177209, 5.66017587502654708850454568150, 5.72722961441655921460060869196, 5.86558213706513342482379828260, 6.13795723302372411961517253201, 6.42027350833503070371002059747
