| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 4·6-s − 4·7-s − 4·8-s + 4·11-s + 4·12-s + 2·13-s + 8·14-s + 3·16-s − 4·17-s − 8·21-s − 8·22-s + 8·23-s − 8·24-s − 4·26-s − 6·27-s − 8·28-s − 4·29-s − 4·31-s + 4·32-s + 8·33-s + 8·34-s − 6·37-s + 4·39-s − 16·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s − 1.51·7-s − 1.41·8-s + 1.20·11-s + 1.15·12-s + 0.554·13-s + 2.13·14-s + 3/4·16-s − 0.970·17-s − 1.74·21-s − 1.70·22-s + 1.66·23-s − 1.63·24-s − 0.784·26-s − 1.15·27-s − 1.51·28-s − 0.742·29-s − 0.718·31-s + 0.707·32-s + 1.39·33-s + 1.37·34-s − 0.986·37-s + 0.640·39-s − 2.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1131791623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1131791623\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + 9 T^{4} + p^{3} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.2.c_c_e_j |
| 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 4 T^{2} - 2 T^{3} + 2 T^{4} - 2 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_e_ac_c |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 102 T^{4} + 36 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.e_m_bk_dy |
| 11 | $\textrm{GL(2,3)}$ | \( 1 - 4 T + 28 T^{2} - 100 T^{3} + 422 T^{4} - 100 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ae_bc_adw_qg |
| 13 | $C_2 \wr S_4$ | \( 1 - 2 T + 28 T^{2} - 46 T^{3} + 410 T^{4} - 46 p T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ac_bc_abu_pu |
| 17 | $C_2 \wr S_4$ | \( 1 + 4 T + 36 T^{2} + 188 T^{3} + 694 T^{4} + 188 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_bk_hg_bas |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 68 T^{2} - 376 T^{3} + 2358 T^{4} - 376 p T^{5} + 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ai_cq_aom_dms |
| 29 | $\textrm{GL(2,3)}$ | \( 1 + 4 T + 84 T^{2} + 332 T^{3} + 3238 T^{4} + 332 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_dg_mu_euo |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 44 T^{2} - 140 T^{3} + 166 T^{4} - 140 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.e_bs_afk_gk |
| 37 | $C_2 \wr S_4$ | \( 1 + 6 T + 124 T^{2} + 626 T^{3} + 6442 T^{4} + 626 p T^{5} + 124 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.g_eu_yc_jnu |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 220 T^{2} + 1936 T^{3} + 14438 T^{4} + 1936 p T^{5} + 220 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.q_im_cwm_vji |
| 43 | $C_2 \wr S_4$ | \( 1 + 4 T + 156 T^{2} + 468 T^{3} + 9750 T^{4} + 468 p T^{5} + 156 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.e_ga_sa_ola |
| 47 | $C_2 \wr S_4$ | \( 1 - 12 T + 124 T^{2} - 1036 T^{3} + 8294 T^{4} - 1036 p T^{5} + 124 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.am_eu_abnw_mha |
| 53 | $C_2 \wr S_4$ | \( 1 + 10 T + 4 p T^{2} + 1406 T^{3} + 16506 T^{4} + 1406 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.k_ie_ccc_ykw |
| 59 | $C_2 \wr S_4$ | \( 1 + 172 T^{2} + 224 T^{3} + 13142 T^{4} + 224 p T^{5} + 172 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_gq_iq_tlm |
| 61 | $C_2 \wr S_4$ | \( 1 - 20 T + 300 T^{2} - 2972 T^{3} + 26502 T^{4} - 2972 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.au_lo_aeki_bnfi |
| 67 | $C_2 \wr S_4$ | \( 1 + 18 T + 276 T^{2} + 3130 T^{3} + 26930 T^{4} + 3130 p T^{5} + 276 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.s_kq_eqk_bnvu |
| 71 | $C_2 \wr S_4$ | \( 1 - 20 T + 316 T^{2} - 3236 T^{3} + 30566 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.au_me_aeum_btfq |
| 73 | $C_2 \wr S_4$ | \( 1 + 28 T + 548 T^{2} + 6916 T^{3} + 69526 T^{4} + 6916 p T^{5} + 548 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.bc_vc_kga_dywc |
| 79 | $C_2 \wr S_4$ | \( 1 - 16 T + 348 T^{2} - 3312 T^{3} + 40646 T^{4} - 3312 p T^{5} + 348 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.aq_nk_aexk_cidi |
| 83 | $C_2 \wr S_4$ | \( 1 + 260 T^{2} + 112 T^{3} + 29862 T^{4} + 112 p T^{5} + 260 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ka_ei_bseo |
| 89 | $C_2 \wr S_4$ | \( 1 + 4 T + 212 T^{2} + 1244 T^{3} + 22134 T^{4} + 1244 p T^{5} + 212 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.e_ie_bvw_bgti |
| 97 | $C_2 \wr S_4$ | \( 1 - 30 T + 612 T^{2} - 8738 T^{3} + 98522 T^{4} - 8738 p T^{5} + 612 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.abe_xo_amyc_fpti |
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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
−5.55831957780802701102929624290, −5.15246460623872295907277896863, −5.00472168803965227868532647747, −4.91919337753455647580120573896, −4.87095283731589626467728724208, −4.40138459379819475665276707866, −4.18511750567391351052589397605, −3.98246666186642075415536015786, −3.77618361448147338352996124393, −3.66701232815744565633755029711, −3.40387793189418260484188304485, −3.38773463347622207573240283077, −3.10702213758783625663503611127, −2.92565380577759537348659903801, −2.87389518637724424294242192621, −2.38718465247247222543359489449, −2.23297413264971271131775167995, −2.19565191718775986648516210973, −1.89984551991789176252924090837, −1.63701869583719858524977034125, −1.26558586567308530625031021253, −1.04507632009281694791297501070, −0.922258182091012672971522706993, −0.25460746240277738723479525534, −0.10128144029075449402628324698,
0.10128144029075449402628324698, 0.25460746240277738723479525534, 0.922258182091012672971522706993, 1.04507632009281694791297501070, 1.26558586567308530625031021253, 1.63701869583719858524977034125, 1.89984551991789176252924090837, 2.19565191718775986648516210973, 2.23297413264971271131775167995, 2.38718465247247222543359489449, 2.87389518637724424294242192621, 2.92565380577759537348659903801, 3.10702213758783625663503611127, 3.38773463347622207573240283077, 3.40387793189418260484188304485, 3.66701232815744565633755029711, 3.77618361448147338352996124393, 3.98246666186642075415536015786, 4.18511750567391351052589397605, 4.40138459379819475665276707866, 4.87095283731589626467728724208, 4.91919337753455647580120573896, 5.00472168803965227868532647747, 5.15246460623872295907277896863, 5.55831957780802701102929624290
