| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 4·7-s + 2·8-s − 4·10-s − 2·13-s − 8·14-s − 8·17-s + 16·19-s − 6·20-s + 4·23-s + 25-s − 4·26-s − 12·28-s + 8·29-s − 20·31-s − 6·32-s − 16·34-s + 8·35-s − 20·37-s + 32·38-s − 4·40-s − 10·43-s + 8·46-s − 12·47-s − 2·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 0.707·8-s − 1.26·10-s − 0.554·13-s − 2.13·14-s − 1.94·17-s + 3.67·19-s − 1.34·20-s + 0.834·23-s + 1/5·25-s − 0.784·26-s − 2.26·28-s + 1.48·29-s − 3.59·31-s − 1.06·32-s − 2.74·34-s + 1.35·35-s − 3.28·37-s + 5.19·38-s − 0.632·40-s − 1.52·43-s + 1.17·46-s − 1.75·47-s − 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4771373317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4771373317\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_b_c_ad |
| 7 | $D_{4}$ | \( ( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.e_s_ce_hv |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_bc_a_qw |
| 13 | $D_4\times C_2$ | \( 1 + 2 T - 15 T^{2} - 14 T^{3} + 140 T^{4} - 14 p T^{5} - 15 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.c_ap_ao_fk |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 22 T^{2} + 64 T^{3} + 387 T^{4} + 64 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.i_w_cm_ox |
| 23 | $C_4\times C_2$ | \( 1 - 4 T - 26 T^{2} + 16 T^{3} + 867 T^{4} + 16 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ae_aba_q_bhj |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 22 T^{2} + 128 T^{3} - 933 T^{4} + 128 p T^{5} + 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ai_w_ey_abjx |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.31.u_ko_dmu_xfn |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.u_kw_dum_bbgl |
| 41 | $C_2^3$ | \( 1 - 74 T^{2} + 3795 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acw_a_fpz |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 3 T^{2} + 170 T^{3} + 4460 T^{4} + 170 p T^{5} - 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.k_ad_go_gpo |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 22 T^{2} + 336 T^{3} + 5907 T^{4} + 336 p T^{5} + 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.m_w_my_itf |
| 53 | $C_2^2$ | \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.e_adq_q_mmt |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aeo_a_plr |
| 61 | $D_4\times C_2$ | \( 1 + 6 T + 33 T^{2} - 714 T^{3} - 5908 T^{4} - 714 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.g_bh_abbm_aitg |
| 67 | $D_4\times C_2$ | \( 1 - 6 T - 35 T^{2} + 378 T^{3} - 1860 T^{4} + 378 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ag_abj_oo_acto |
| 71 | $C_2^2$ | \( ( 1 - 10 T + 29 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.au_gc_acyy_bleh |
| 73 | $D_4\times C_2$ | \( 1 + 2 T - 71 T^{2} - 142 T^{3} + 4 T^{4} - 142 p T^{5} - 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.c_act_afm_e |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.as_en_abhy_qho |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.83.abg_bbo_aovg_gfna |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 448 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ai_acg_rg_bwt |
| 97 | $C_2^2$ | \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.am_adi_aqq_brrn |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.98640162379484144943588917881, −6.98427089782899625206182493341, −6.78177968712170771112649359570, −6.72688556460919425717174805658, −6.40476831805804803909153137550, −6.40437767461734359444151609195, −5.71111025177151462283641598065, −5.55215288811809189456164127551, −5.47604792205812285833211971385, −5.07077938489908334972766734758, −4.91284907153272873165746223580, −4.89282408308823625802978593287, −4.66718936476540772382927057986, −3.92471164930328643949208812155, −3.83131101300964818251869368282, −3.47933169873033164158730829236, −3.43197614292778025536600580339, −3.25065695249925714829790988553, −3.05356233117805522530622085727, −2.77931632868215756296614681487, −2.05070944851746593685893097010, −1.99465804068466328322066550493, −1.65939686926094256625508042628, −0.918218696833690515175329225242, −0.14091225595846314830869586172,
0.14091225595846314830869586172, 0.918218696833690515175329225242, 1.65939686926094256625508042628, 1.99465804068466328322066550493, 2.05070944851746593685893097010, 2.77931632868215756296614681487, 3.05356233117805522530622085727, 3.25065695249925714829790988553, 3.43197614292778025536600580339, 3.47933169873033164158730829236, 3.83131101300964818251869368282, 3.92471164930328643949208812155, 4.66718936476540772382927057986, 4.89282408308823625802978593287, 4.91284907153272873165746223580, 5.07077938489908334972766734758, 5.47604792205812285833211971385, 5.55215288811809189456164127551, 5.71111025177151462283641598065, 6.40437767461734359444151609195, 6.40476831805804803909153137550, 6.72688556460919425717174805658, 6.78177968712170771112649359570, 6.98427089782899625206182493341, 6.98640162379484144943588917881
