| L(s) = 1 | + 3·2-s + 3-s + 7·4-s − 8·5-s + 3·6-s + 3·7-s + 15·8-s − 24·10-s − 13·11-s + 7·12-s + 5·13-s + 9·14-s − 8·15-s + 30·16-s + 14·19-s − 56·20-s + 3·21-s − 39·22-s + 4·23-s + 15·24-s + 35·25-s + 15·26-s + 21·28-s + 16·29-s − 24·30-s − 13·31-s + 57·32-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 7/2·4-s − 3.57·5-s + 1.22·6-s + 1.13·7-s + 5.30·8-s − 7.58·10-s − 3.91·11-s + 2.02·12-s + 1.38·13-s + 2.40·14-s − 2.06·15-s + 15/2·16-s + 3.21·19-s − 12.5·20-s + 0.654·21-s − 8.31·22-s + 0.834·23-s + 3.06·24-s + 7·25-s + 2.94·26-s + 3.96·28-s + 2.97·29-s − 4.38·30-s − 2.33·31-s + 10.0·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 71^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 71^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.023088791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.023088791\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) | |
| 71 | $C_4$ | \( 1 + 19 T + 201 T^{2} + 19 p T^{3} + p^{2} T^{4} \) | |
| good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.2.ad_c_a_b |
| 5 | $C_2^2:C_4$ | \( 1 + 8 T + 29 T^{2} + 72 T^{3} + 161 T^{4} + 72 p T^{5} + 29 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.i_bd_cu_gf |
| 7 | $C_2^2:C_4$ | \( 1 - 3 T - 3 T^{2} + 5 T^{3} + 36 T^{4} + 5 p T^{5} - 3 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ad_ad_f_bk |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) | 4.11.n_df_nx_bxg |
| 13 | $C_4\times C_2$ | \( 1 - 5 T + 2 T^{2} - 25 T^{3} + 259 T^{4} - 25 p T^{5} + 2 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.af_c_az_jz |
| 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 | $C_2^2:C_4$ | \( 1 - 14 T + 77 T^{2} - 262 T^{3} + 955 T^{4} - 262 p T^{5} + 77 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ao_cz_akc_bkt |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_cg_ahs_cxv |
| 29 | $C_4\times C_2$ | \( 1 - 16 T + 157 T^{2} - 1208 T^{3} + 7425 T^{4} - 1208 p T^{5} + 157 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.aq_gb_abum_kzp |
| 31 | $C_2^2:C_4$ | \( 1 + 13 T + 38 T^{2} - 109 T^{3} - 995 T^{4} - 109 p T^{5} + 38 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.n_bm_aef_abmh |
| 37 | $D_{4}$ | \( ( 1 + 7 T + 85 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.o_il_cns_ucr |
| 41 | $D_{4}$ | \( ( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.k_fj_bls_mvd |
| 43 | $C_2^2:C_4$ | \( 1 - 4 T - 37 T^{2} + 130 T^{3} + 1291 T^{4} + 130 p T^{5} - 37 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_abl_fa_bxr |
| 47 | $C_4\times C_2$ | \( 1 + 15 T + 88 T^{2} + 675 T^{3} + 6349 T^{4} + 675 p T^{5} + 88 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.p_dk_zz_jkf |
| 53 | $C_2^2:C_4$ | \( 1 - 9 T + 83 T^{2} - 615 T^{3} + 2056 T^{4} - 615 p T^{5} + 83 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.aj_df_axr_dbc |
| 59 | $C_2^2:C_4$ | \( 1 - 5 T - 19 T^{2} - 345 T^{3} + 5276 T^{4} - 345 p T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.af_at_anh_huy |
| 61 | $C_2^2:C_4$ | \( 1 + 5 T - T^{2} - 365 T^{3} - 1184 T^{4} - 365 p T^{5} - p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.f_ab_aob_abto |
| 67 | $C_2^2:C_4$ | \( 1 - 16 T + 29 T^{2} + 868 T^{3} - 9191 T^{4} + 868 p T^{5} + 29 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.aq_bd_bhk_anpn |
| 73 | $C_2^2:C_4$ | \( 1 - 20 T + 117 T^{2} - 460 T^{3} + 5069 T^{4} - 460 p T^{5} + 117 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.au_en_ars_hmz |
| 79 | $C_2^2:C_4$ | \( 1 + 27 T + 210 T^{2} - 703 T^{3} - 20211 T^{4} - 703 p T^{5} + 210 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.bb_ic_abbb_abdxj |
| 83 | $C_4\times C_2$ | \( 1 + 17 T + 2 p T^{2} + 2131 T^{3} + 26289 T^{4} + 2131 p T^{5} + 2 p^{3} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.r_gk_ddz_bmxd |
| 89 | $C_2^2:C_4$ | \( 1 - 14 T - 13 T^{2} + 1188 T^{3} - 10075 T^{4} + 1188 p T^{5} - 13 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ao_an_bts_aoxn |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.am_kw_adxo_cioh |
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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
−8.657504294514686496638575839232, −8.468254994249665063250251959930, −8.112208082405362211383533337807, −8.096227958412400665959751705715, −8.081405341052435696004478041412, −7.59972062186585577381821141652, −7.31032489314366304360339425940, −7.14622723332065673927687494759, −7.13050263143323316425028474360, −6.99038627652996891295547043485, −6.08664674409370473602490251731, −5.81486769009499537336202101818, −5.37040867994625547979770983517, −5.14734145331997360538862683545, −5.00584740620960110484472987836, −4.70565295925616113722179114199, −4.69687698929824595521888919286, −3.85489758778458500192809170725, −3.76442404849835522318726638892, −3.29982979165322263147623790740, −3.25117897537460985134727500105, −2.88436744098542462967461381731, −2.62845813300962637405883249060, −1.74113079865308311380341710844, −0.983928934806867949558468569756,
0.983928934806867949558468569756, 1.74113079865308311380341710844, 2.62845813300962637405883249060, 2.88436744098542462967461381731, 3.25117897537460985134727500105, 3.29982979165322263147623790740, 3.76442404849835522318726638892, 3.85489758778458500192809170725, 4.69687698929824595521888919286, 4.70565295925616113722179114199, 5.00584740620960110484472987836, 5.14734145331997360538862683545, 5.37040867994625547979770983517, 5.81486769009499537336202101818, 6.08664674409370473602490251731, 6.99038627652996891295547043485, 7.13050263143323316425028474360, 7.14622723332065673927687494759, 7.31032489314366304360339425940, 7.59972062186585577381821141652, 8.081405341052435696004478041412, 8.096227958412400665959751705715, 8.112208082405362211383533337807, 8.468254994249665063250251959930, 8.657504294514686496638575839232
