| L(s) = 1 | − 4·5-s − 2·7-s + 4·11-s + 4·13-s − 8·17-s − 4·19-s + 4·23-s − 25-s − 4·29-s + 8·35-s + 8·37-s − 16·41-s − 6·43-s − 13·49-s − 16·53-s − 16·55-s − 4·59-s + 6·61-s − 16·65-s − 8·67-s + 12·71-s + 6·73-s − 8·77-s + 20·79-s − 8·83-s + 32·85-s − 32·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 1.20·11-s + 1.10·13-s − 1.94·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.742·29-s + 1.35·35-s + 1.31·37-s − 2.49·41-s − 0.914·43-s − 1.85·49-s − 2.19·53-s − 2.15·55-s − 0.520·59-s + 0.768·61-s − 1.98·65-s − 0.977·67-s + 1.42·71-s + 0.702·73-s − 0.911·77-s + 2.25·79-s − 0.878·83-s + 3.47·85-s − 3.39·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 4 T + 17 T^{2} + 48 T^{3} + 128 T^{4} + 48 p T^{5} + 17 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.e_r_bw_ey |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 T + 17 T^{2} + 26 T^{3} + 156 T^{4} + 26 p T^{5} + 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.c_r_ba_ga |
| 11 | $C_2 \wr S_4$ | \( 1 - 4 T + 41 T^{2} - 116 T^{3} + 664 T^{4} - 116 p T^{5} + 41 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ae_bp_aem_zo |
| 13 | $C_2 \wr S_4$ | \( 1 - 4 T + 32 T^{2} - 76 T^{3} + 462 T^{4} - 76 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_bg_acy_ru |
| 17 | $C_2 \wr S_4$ | \( 1 + 8 T + 65 T^{2} + 336 T^{3} + 1712 T^{4} + 336 p T^{5} + 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.i_cn_my_cnw |
| 23 | $C_2 \wr S_4$ | \( 1 - 4 T + 36 T^{2} - 228 T^{3} + 614 T^{4} - 228 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ae_bk_aiu_xq |
| 29 | $C_2 \wr S_4$ | \( 1 + 4 T + 96 T^{2} + 268 T^{3} + 3854 T^{4} + 268 p T^{5} + 96 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_ds_ki_fsg |
| 31 | $C_2 \wr S_4$ | \( 1 + 56 T^{2} + 176 T^{3} + 1710 T^{4} + 176 p T^{5} + 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ce_gu_cnu |
| 37 | $C_2 \wr S_4$ | \( 1 - 8 T + 104 T^{2} - 472 T^{3} + 4510 T^{4} - 472 p T^{5} + 104 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_ea_ase_grm |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 152 T^{2} + 816 T^{3} + 4814 T^{4} + 816 p T^{5} + 152 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.q_fw_bfk_hde |
| 43 | $C_2 \wr S_4$ | \( 1 + 6 T + 65 T^{2} + 270 T^{3} + 2740 T^{4} + 270 p T^{5} + 65 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.g_cn_kk_ebk |
| 47 | $C_2 \wr S_4$ | \( 1 + 161 T^{2} + 4 T^{3} + 10880 T^{4} + 4 p T^{5} + 161 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_gf_e_qcm |
| 53 | $C_2 \wr S_4$ | \( 1 + 16 T + 236 T^{2} + 2352 T^{3} + 19478 T^{4} + 2352 p T^{5} + 236 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.q_jc_dmm_bcve |
| 59 | $C_2 \wr S_4$ | \( 1 + 4 T + 96 T^{2} - 444 T^{3} + 2126 T^{4} - 444 p T^{5} + 96 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.e_ds_arc_ddu |
| 61 | $C_2 \wr S_4$ | \( 1 - 6 T + 121 T^{2} - 766 T^{3} + 11180 T^{4} - 766 p T^{5} + 121 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ag_er_abdm_qoa |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T + 256 T^{2} + 1480 T^{3} + 25390 T^{4} + 1480 p T^{5} + 256 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.i_jw_cey_bloo |
| 71 | $C_2 \wr S_4$ | \( 1 - 12 T + 288 T^{2} - 2396 T^{3} + 30878 T^{4} - 2396 p T^{5} + 288 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.am_lc_adoe_btrq |
| 73 | $C_2^3:S_4$ | \( 1 - 6 T + 17 T^{2} + 354 T^{3} + 1204 T^{4} + 354 p T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ag_r_nq_bui |
| 79 | $C_2 \wr S_4$ | \( 1 - 20 T + 264 T^{2} - 3316 T^{3} + 34254 T^{4} - 3316 p T^{5} + 264 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.au_ke_aexo_byrm |
| 83 | $C_2 \wr S_4$ | \( 1 + 8 T + 84 T^{2} - 152 T^{3} - 1098 T^{4} - 152 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.i_dg_afw_abqg |
| 89 | $C_2 \wr S_4$ | \( 1 + 32 T + 668 T^{2} + 9568 T^{3} + 104038 T^{4} + 9568 p T^{5} + 668 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.bg_zs_oea_fxxm |
| 97 | $C_2 \wr S_4$ | \( 1 + 352 T^{2} - 16 T^{3} + 49694 T^{4} - 16 p T^{5} + 352 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_no_aq_cvni |
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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
−6.28779832377612536259285953635, −5.95097285193104147159656199617, −5.75915265091369956474227492066, −5.67439905168931042948545154925, −5.36010972723958936688569261193, −4.90005326109565253461627835688, −4.89864065586000273867645797408, −4.87151842386648343947914183147, −4.64145869803153344360080361367, −4.09462819893594953386359964417, −4.05570053234411123559752060344, −4.05443081330152084389111925015, −3.99964119951767926339498035810, −3.46973933152056898106794786785, −3.45760425741865374046104190609, −3.29857014435694258245620203710, −3.18838876642273856157751143565, −2.63246509019476905573581718526, −2.45893949992380783055969102557, −2.41159818787717882335141921641, −1.93898215628925007096266456209, −1.73755377199162604085378359740, −1.34480634974665864338029734680, −1.20738908948628920764316092686, −1.11308026720804679875431357605, 0, 0, 0, 0,
1.11308026720804679875431357605, 1.20738908948628920764316092686, 1.34480634974665864338029734680, 1.73755377199162604085378359740, 1.93898215628925007096266456209, 2.41159818787717882335141921641, 2.45893949992380783055969102557, 2.63246509019476905573581718526, 3.18838876642273856157751143565, 3.29857014435694258245620203710, 3.45760425741865374046104190609, 3.46973933152056898106794786785, 3.99964119951767926339498035810, 4.05443081330152084389111925015, 4.05570053234411123559752060344, 4.09462819893594953386359964417, 4.64145869803153344360080361367, 4.87151842386648343947914183147, 4.89864065586000273867645797408, 4.90005326109565253461627835688, 5.36010972723958936688569261193, 5.67439905168931042948545154925, 5.75915265091369956474227492066, 5.95097285193104147159656199617, 6.28779832377612536259285953635
