| L(s) = 1 | − 6·9-s − 6·11-s + 2·19-s + 2·29-s + 2·31-s + 30·41-s − 2·59-s + 24·61-s + 12·71-s + 14·79-s + 9·81-s + 28·89-s + 36·99-s + 24·101-s − 16·109-s + 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s − 12·171-s + ⋯ |
| L(s) = 1 | − 2·9-s − 1.80·11-s + 0.458·19-s + 0.371·29-s + 0.359·31-s + 4.68·41-s − 0.260·59-s + 3.07·61-s + 1.42·71-s + 1.57·79-s + 81-s + 2.96·89-s + 3.61·99-s + 2.38·101-s − 1.53·109-s + 7/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s − 0.917·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.431453175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.431453175\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_g_a_bb |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.g_bd_ew_uu |
| 13 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_i_a_ew |
| 17 | $D_4\times C_2$ | \( 1 + 45 T^{2} + 956 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_bt_a_bku |
| 19 | $D_{4}$ | \( ( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ac_bx_adi_bzk |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_be_a_oh |
| 29 | $D_{4}$ | \( ( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ac_dl_afq_flk |
| 31 | $D_{4}$ | \( ( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ac_dt_agc_giy |
| 37 | $D_4\times C_2$ | \( 1 + 17 T^{2} + 1656 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_r_a_cls |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.abe_sf_ahik_cdai |
| 43 | $D_4\times C_2$ | \( 1 + 125 T^{2} + 7248 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_ev_a_ksu |
| 47 | $D_4\times C_2$ | \( 1 + 3 p T^{2} + 9032 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) | 4.47.a_fl_a_njk |
| 53 | $D_4\times C_2$ | \( 1 + 125 T^{2} + 9396 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ev_a_nxk |
| 59 | $D_{4}$ | \( ( 1 + T + 104 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.c_ib_mo_bami |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.ay_ni_afto_cach |
| 67 | $D_4\times C_2$ | \( 1 + 62 T^{2} + 1731 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ck_a_cop |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.am_iq_acye_bjog |
| 73 | $D_4\times C_2$ | \( 1 + 245 T^{2} + 25308 T^{4} + 245 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_jl_a_bllk |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 156 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ao_nx_aewo_cnxw |
| 83 | $D_4\times C_2$ | \( 1 + 245 T^{2} + 28656 T^{4} + 245 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_jl_a_bqke |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) | 4.89.abc_za_anci_fvhb |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_lg_a_chjq |
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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
−6.01386146689088099326891195109, −5.49029977667496697965037153868, −5.36470254792710437397608647081, −5.33678476938110887404819960419, −5.20570894903667854883256791398, −4.93572503787074097524276285394, −4.62148713867238688928512518217, −4.47715449527884593327618765704, −4.44899657437765326272886649768, −3.84563374900435508139017491107, −3.84488628235524053666349281441, −3.59045576764620882413256763925, −3.57392938751520457393356303725, −2.97452870718386812431410441457, −2.83297548575529965345047392015, −2.74077712759499915437497592392, −2.73959997294310563654271200821, −2.28591405637713447536171358815, −2.16426371358390404195355808420, −1.96667246598051312791277239387, −1.64241022631527792798759215330, −0.873968337279449952461044586681, −0.812578734995134305732650201349, −0.70924900676691445367596642643, −0.32561725763582843796863911732,
0.32561725763582843796863911732, 0.70924900676691445367596642643, 0.812578734995134305732650201349, 0.873968337279449952461044586681, 1.64241022631527792798759215330, 1.96667246598051312791277239387, 2.16426371358390404195355808420, 2.28591405637713447536171358815, 2.73959997294310563654271200821, 2.74077712759499915437497592392, 2.83297548575529965345047392015, 2.97452870718386812431410441457, 3.57392938751520457393356303725, 3.59045576764620882413256763925, 3.84488628235524053666349281441, 3.84563374900435508139017491107, 4.44899657437765326272886649768, 4.47715449527884593327618765704, 4.62148713867238688928512518217, 4.93572503787074097524276285394, 5.20570894903667854883256791398, 5.33678476938110887404819960419, 5.36470254792710437397608647081, 5.49029977667496697965037153868, 6.01386146689088099326891195109
