| L(s) = 1 | + 2·5-s − 5·9-s + 8·17-s − 9·25-s − 20·29-s + 8·37-s − 10·45-s − 5·49-s − 2·53-s − 12·61-s + 42·73-s + 5·81-s + 16·85-s − 18·89-s − 2·97-s + 16·101-s + 12·109-s + 16·113-s − 13·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 5/3·9-s + 1.94·17-s − 9/5·25-s − 3.71·29-s + 1.31·37-s − 1.49·45-s − 5/7·49-s − 0.274·53-s − 1.53·61-s + 4.91·73-s + 5/9·81-s + 1.73·85-s − 1.90·89-s − 0.203·97-s + 1.59·101-s + 1.14·109-s + 1.50·113-s − 1.18·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.808314280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.808314280\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | | \( 1 \) | |
| good | 3 | $C_2^2:C_4$ | \( 1 + 5 T^{2} + 20 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_f_a_u |
| 5 | $D_{4}$ | \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ac_n_aw_ds |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 5 T^{2} + 100 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_f_a_dw |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 13 T^{2} + 76 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_n_a_cy |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_cg_als_cid |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 3 T^{2} + 516 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_ad_a_tw |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 29 T^{2} + 924 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bd_a_bjo |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.29.u_kg_die_vgt |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 2806 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_cq_a_edy |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ai_fi_abee_lhz |
| 41 | $C_2^2$ | \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_fa_a_lfv |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 101 T^{2} + 5020 T^{4} + 101 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_dx_a_hlc |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 72 T^{2} + 2382 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_cu_a_dnq |
| 53 | $D_{4}$ | \( ( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.c_hx_ly_xwi |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 21 T^{2} + 7068 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_v_a_klw |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.m_gg_cfg_xjv |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 149 T^{2} + 13300 T^{4} + 149 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ft_a_tro |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 205 T^{2} + 20380 T^{4} + 205 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_hx_a_bedw |
| 73 | $D_{4}$ | \( ( 1 - 21 T + 218 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.abq_bht_ascc_gzie |
| 79 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 10174 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_acm_a_pbi |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 252 T^{2} + 28566 T^{4} + 252 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_js_a_bqgs |
| 89 | $D_{4}$ | \( ( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.s_pl_gqk_deqm |
| 97 | $D_{4}$ | \( ( 1 + T + 88 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.c_gv_og_bnpc |
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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
−5.83929581889116781960108422935, −5.57912235069714940301745037323, −5.37638629793343289107260941343, −5.34639300502450257772973104442, −5.18006601446849506787489692288, −4.82574071743881579282654378933, −4.65426028802467556753714853989, −4.43283834564460160763679492543, −4.14475027816488663214398999224, −3.91595699164119802455057400621, −3.61965786381508758424748331480, −3.56425661205340383173181979744, −3.54276880073823827291097436863, −3.14373972485910045100165482621, −2.88533637917719291020499998240, −2.68968919614333288728510250930, −2.66543677917204874676476810142, −1.98110309694714902692313067561, −1.96351248886836598003606993016, −1.88973696696187482356225663722, −1.69071349775762249863402761818, −1.34919484456717199763626295918, −0.67253584796936636256325940460, −0.57456093185557302966688042000, −0.38803538483078496095530353821,
0.38803538483078496095530353821, 0.57456093185557302966688042000, 0.67253584796936636256325940460, 1.34919484456717199763626295918, 1.69071349775762249863402761818, 1.88973696696187482356225663722, 1.96351248886836598003606993016, 1.98110309694714902692313067561, 2.66543677917204874676476810142, 2.68968919614333288728510250930, 2.88533637917719291020499998240, 3.14373972485910045100165482621, 3.54276880073823827291097436863, 3.56425661205340383173181979744, 3.61965786381508758424748331480, 3.91595699164119802455057400621, 4.14475027816488663214398999224, 4.43283834564460160763679492543, 4.65426028802467556753714853989, 4.82574071743881579282654378933, 5.18006601446849506787489692288, 5.34639300502450257772973104442, 5.37638629793343289107260941343, 5.57912235069714940301745037323, 5.83929581889116781960108422935
