| L(s) = 1 | − 2-s + 3·3-s + 2·4-s − 5-s − 3·6-s − 3·7-s + 3·9-s + 10-s + 6·12-s + 4·13-s + 3·14-s − 3·15-s − 3·18-s + 4·19-s − 2·20-s − 9·21-s − 32·23-s − 4·26-s − 6·28-s + 6·29-s + 3·30-s + 2·31-s + 11·32-s + 3·35-s + 6·36-s + 8·37-s − 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 4-s − 0.447·5-s − 1.22·6-s − 1.13·7-s + 9-s + 0.316·10-s + 1.73·12-s + 1.10·13-s + 0.801·14-s − 0.774·15-s − 0.707·18-s + 0.917·19-s − 0.447·20-s − 1.96·21-s − 6.67·23-s − 0.784·26-s − 1.13·28-s + 1.11·29-s + 0.547·30-s + 0.359·31-s + 1.94·32-s + 0.507·35-s + 36-s + 1.31·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9741361694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9741361694\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_4\times C_2$ | \( 1 + T - T^{2} - 3 T^{3} - T^{4} - 3 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.2.b_ab_ad_ab |
| 3 | $C_4\times C_2$ | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.3.ad_g_aj_j |
| 7 | $C_4\times C_2$ | \( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 15 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.d_c_ap_ach |
| 13 | $C_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 40 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_d_bo_ahr |
| 17 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.17.a_ar_a_ld |
| 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.ae_ad_dk_alj |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.23.bg_si_ghs_bkxa |
| 29 | $C_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 132 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ag_h_fc_abmh |
| 31 | $C_4\times C_2$ | \( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 116 p T^{5} - 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ac_abb_em_xh |
| 37 | $C_4\times C_2$ | \( 1 - 8 T + 27 T^{2} + 80 T^{3} - 1639 T^{4} + 80 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_bb_dc_aclb |
| 41 | $C_4\times C_2$ | \( 1 + 5 T - 16 T^{2} - 285 T^{3} - 769 T^{4} - 285 p T^{5} - 16 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.f_aq_akz_abdp |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.43.u_mk_eom_bkkx |
| 47 | $C_4\times C_2$ | \( 1 - 3 T - 38 T^{2} + 255 T^{3} + 1021 T^{4} + 255 p T^{5} - 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ad_abm_jv_bnh |
| 53 | $C_4\times C_2$ | \( 1 + 4 T - 37 T^{2} - 360 T^{3} + 521 T^{4} - 360 p T^{5} - 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.e_abl_anw_ub |
| 59 | $C_4\times C_2$ | \( 1 - 2 T - 55 T^{2} + 228 T^{3} + 2789 T^{4} + 228 p T^{5} - 55 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ac_acd_iu_edh |
| 61 | $C_4\times C_2$ | \( 1 + 11 T + 60 T^{2} - 11 T^{3} - 3781 T^{4} - 11 p T^{5} + 60 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.l_ci_al_afpl |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) | 4.67.ca_bxi_bcma_kxcl |
| 71 | $C_4\times C_2$ | \( 1 + 2 T - 67 T^{2} - 276 T^{3} + 4205 T^{4} - 276 p T^{5} - 67 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.c_acp_akq_gft |
| 73 | $C_4\times C_2$ | \( 1 + 8 T - 9 T^{2} - 656 T^{3} - 4591 T^{4} - 656 p T^{5} - 9 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.i_aj_azg_agup |
| 79 | $C_4\times C_2$ | \( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 580 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ak_v_wi_alax |
| 83 | $C_4\times C_2$ | \( 1 - 4 T - 67 T^{2} + 600 T^{3} + 3161 T^{4} + 600 p T^{5} - 67 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ae_acp_xc_erp |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.89.ae_ny_abpg_ctxb |
| 97 | $C_4\times C_2$ | \( 1 - 8 T - 33 T^{2} + 1040 T^{3} - 5119 T^{4} + 1040 p T^{5} - 33 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ai_abh_boa_ahox |
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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
−7.68526877783458420497032390540, −7.64343735898290953660201753794, −7.62238297446304699059498245615, −6.96791076227146943077935863078, −6.79576398921989384411615240376, −6.45367993122772090489773046504, −6.16547955119399647942857928857, −6.10462450170144042641760949240, −6.07876535086056119754467285820, −5.70677621803806252157829683942, −5.43611200995842551555214591546, −4.62321145103935379412868272186, −4.55671252444591504234826754666, −4.32204418098394331230589003818, −4.21097683628254417425972646215, −3.68517248052460896578672336191, −3.31137189273840002909054879842, −3.28640092794010035492247148150, −3.14380066441807536764370917132, −2.58991243857458834174726776844, −2.20741332072551306284384933886, −2.09322317775140740705668739360, −1.49866035777628387405669385230, −1.43357776583375152948169618928, −0.26532395216767689579197965999,
0.26532395216767689579197965999, 1.43357776583375152948169618928, 1.49866035777628387405669385230, 2.09322317775140740705668739360, 2.20741332072551306284384933886, 2.58991243857458834174726776844, 3.14380066441807536764370917132, 3.28640092794010035492247148150, 3.31137189273840002909054879842, 3.68517248052460896578672336191, 4.21097683628254417425972646215, 4.32204418098394331230589003818, 4.55671252444591504234826754666, 4.62321145103935379412868272186, 5.43611200995842551555214591546, 5.70677621803806252157829683942, 6.07876535086056119754467285820, 6.10462450170144042641760949240, 6.16547955119399647942857928857, 6.45367993122772090489773046504, 6.79576398921989384411615240376, 6.96791076227146943077935863078, 7.62238297446304699059498245615, 7.64343735898290953660201753794, 7.68526877783458420497032390540
