| L(s) = 1 | − 2·2-s + 2·3-s + 4-s + 5·5-s − 4·6-s − 4·7-s + 2·8-s + 2·9-s − 10·10-s − 4·11-s + 2·12-s + 5·13-s + 8·14-s + 10·15-s − 4·16-s + 9·17-s − 4·18-s + 5·20-s − 8·21-s + 8·22-s − 6·23-s + 4·24-s + 15·25-s − 10·26-s − 8·27-s − 4·28-s − 7·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 2.23·5-s − 1.63·6-s − 1.51·7-s + 0.707·8-s + 2/3·9-s − 3.16·10-s − 1.20·11-s + 0.577·12-s + 1.38·13-s + 2.13·14-s + 2.58·15-s − 16-s + 2.18·17-s − 0.942·18-s + 1.11·20-s − 1.74·21-s + 1.70·22-s − 1.25·23-s + 0.816·24-s + 3·25-s − 1.96·26-s − 1.53·27-s − 0.755·28-s − 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.010436229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010436229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 19 | | \( 1 \) | |
| good | 3 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_c_i_ar |
| 5 | $C_4\times C_2$ | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + 3 p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.5.af_k_az_cx |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.e_y_cq_ju |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.e_bo_em_ze |
| 13 | $D_4\times C_2$ | \( 1 - 5 T - 6 T^{2} - 25 T^{3} + 467 T^{4} - 25 p T^{5} - 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.af_ag_az_rz |
| 17 | $D_4\times C_2$ | \( 1 - 9 T + 28 T^{2} - 171 T^{3} + 1143 T^{4} - 171 p T^{5} + 28 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.aj_bc_agp_brz |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 14 T^{2} + 24 T^{3} + 1143 T^{4} + 24 p T^{5} - 14 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.g_ao_y_brz |
| 29 | $D_4\times C_2$ | \( 1 + 7 T + 10 T^{2} - 133 T^{3} - 741 T^{4} - 133 p T^{5} + 10 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.h_k_afd_abcn |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.e_eq_ns_iew |
| 37 | $D_{4}$ | \( ( 1 + 19 T + 163 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.bm_bal_lgi_dewr |
| 41 | $D_4\times C_2$ | \( 1 + 5 T - 52 T^{2} - 25 T^{3} + 3223 T^{4} - 25 p T^{5} - 52 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.f_aca_az_etz |
| 43 | $D_4\times C_2$ | \( 1 - 14 T + 66 T^{2} - 616 T^{3} + 6623 T^{4} - 616 p T^{5} + 66 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ao_co_axs_jut |
| 47 | $C_2^3$ | \( 1 - 74 T^{2} + 3267 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_acw_a_evr |
| 53 | $D_4\times C_2$ | \( 1 + T - 74 T^{2} - 31 T^{3} + 2763 T^{4} - 31 p T^{5} - 74 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.53.b_acw_abf_ech |
| 59 | $C_4\times C_2$ | \( 1 - 8 T - 50 T^{2} + 32 T^{3} + 5739 T^{4} + 32 p T^{5} - 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ai_aby_bg_imt |
| 61 | $D_4\times C_2$ | \( 1 + 5 T - 102 T^{2} + 25 T^{3} + 10883 T^{4} + 25 p T^{5} - 102 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.f_ady_z_qcp |
| 67 | $D_4\times C_2$ | \( 1 - 10 T - 14 T^{2} + 200 T^{3} + 1807 T^{4} + 200 p T^{5} - 14 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ak_ao_hs_crn |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 46 T^{2} - 528 T^{3} - 5661 T^{4} - 528 p T^{5} + 46 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.m_bu_aui_aijt |
| 73 | $D_4\times C_2$ | \( 1 - 9 T - 84 T^{2} - 171 T^{3} + 15983 T^{4} - 171 p T^{5} - 84 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.aj_adg_agp_xqt |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 78 T^{2} + 200 T^{3} + 17543 T^{4} + 200 p T^{5} - 78 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.k_ada_hs_zyt |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.aq_oy_afuu_cwxy |
| 89 | $D_4\times C_2$ | \( 1 - 11 T - 76 T^{2} - 209 T^{3} + 20119 T^{4} - 209 p T^{5} - 76 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.al_acy_aib_bdtv |
| 97 | $D_4\times C_2$ | \( 1 - 15 T - 24 T^{2} - 825 T^{3} + 30767 T^{4} - 825 p T^{5} - 24 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ap_ay_abft_btnj |
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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
−7.56770578914605770805297710412, −7.48718246076117438796055171904, −7.05177637006031362485622541432, −7.00164771604062078997721534675, −6.52001672390653700584986518904, −6.29308833631848726773915605076, −6.08800207767426882333125787534, −5.91792496657641982315668487452, −5.58070517628957070579738600504, −5.55914904802321593811956206939, −5.08452195503890843772970455460, −4.99643510671926578728302809580, −4.87640257880231938157291590842, −4.00816511215324719900816029028, −3.77048739167155032905645609999, −3.55890543934995366841479222479, −3.40525539001598431616560734667, −3.22031965290119469632352592551, −2.96079781071457665273803340155, −2.17520127326718198609792065861, −1.98269953821665039382745306698, −1.93638475503339328773278316427, −1.67619233084467661411277240123, −1.08353751601116611642036280093, −0.31536143006747476238841005996,
0.31536143006747476238841005996, 1.08353751601116611642036280093, 1.67619233084467661411277240123, 1.93638475503339328773278316427, 1.98269953821665039382745306698, 2.17520127326718198609792065861, 2.96079781071457665273803340155, 3.22031965290119469632352592551, 3.40525539001598431616560734667, 3.55890543934995366841479222479, 3.77048739167155032905645609999, 4.00816511215324719900816029028, 4.87640257880231938157291590842, 4.99643510671926578728302809580, 5.08452195503890843772970455460, 5.55914904802321593811956206939, 5.58070517628957070579738600504, 5.91792496657641982315668487452, 6.08800207767426882333125787534, 6.29308833631848726773915605076, 6.52001672390653700584986518904, 7.00164771604062078997721534675, 7.05177637006031362485622541432, 7.48718246076117438796055171904, 7.56770578914605770805297710412
