| L(s) = 1 | − 3·3-s − 5·5-s − 11·7-s + 5·11-s + 4·13-s + 15·15-s + 4·19-s + 33·21-s − 4·23-s + 15·25-s + 11·27-s − 4·29-s − 8·31-s − 15·33-s + 55·35-s + 5·37-s − 12·39-s − 5·41-s − 10·43-s − 7·47-s + 54·49-s − 53-s − 25·55-s − 12·57-s + 30·59-s − 14·61-s − 20·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.23·5-s − 4.15·7-s + 1.50·11-s + 1.10·13-s + 3.87·15-s + 0.917·19-s + 7.20·21-s − 0.834·23-s + 3·25-s + 2.11·27-s − 0.742·29-s − 1.43·31-s − 2.61·33-s + 9.29·35-s + 0.821·37-s − 1.92·39-s − 0.780·41-s − 1.52·43-s − 1.02·47-s + 54/7·49-s − 0.137·53-s − 3.37·55-s − 1.58·57-s + 3.90·59-s − 1.79·61-s − 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 37 | $C_1$ | \( ( 1 - T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 + p T + p^{2} T^{2} + 16 T^{3} + 40 T^{4} + 64 T^{5} + 40 p T^{6} + 16 p^{2} T^{7} + p^{5} T^{8} + p^{5} T^{9} + p^{5} T^{10} \) | 5.3.d_j_q_bo_cm |
| 7 | $C_2 \wr S_5$ | \( 1 + 11 T + 67 T^{2} + 276 T^{3} + 894 T^{4} + 2484 T^{5} + 894 p T^{6} + 276 p^{2} T^{7} + 67 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.l_cp_kq_bik_dro |
| 11 | $C_2 \wr S_5$ | \( 1 - 5 T + 47 T^{2} - 172 T^{3} + 962 T^{4} - 2670 T^{5} + 962 p T^{6} - 172 p^{2} T^{7} + 47 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.af_bv_agq_bla_adys |
| 13 | $C_2 \wr S_5$ | \( 1 - 4 T + 37 T^{2} - 148 T^{3} + 746 T^{4} - 2752 T^{5} + 746 p T^{6} - 148 p^{2} T^{7} + 37 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.ae_bl_afs_bcs_aebw |
| 17 | $C_2 \wr S_5$ | \( 1 + 33 T^{2} + 12 T^{3} + 594 T^{4} + 600 T^{5} + 594 p T^{6} + 12 p^{2} T^{7} + 33 p^{3} T^{8} + p^{5} T^{10} \) | 5.17.a_bh_m_ww_xc |
| 19 | $C_2 \wr S_5$ | \( 1 - 4 T + 3 p T^{2} - 238 T^{3} + 1522 T^{4} - 6148 T^{5} + 1522 p T^{6} - 238 p^{2} T^{7} + 3 p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.ae_cf_aje_cgo_ajcm |
| 23 | $C_2 \wr S_5$ | \( 1 + 4 T + 43 T^{2} + 160 T^{3} + 1106 T^{4} + 2936 T^{5} + 1106 p T^{6} + 160 p^{2} T^{7} + 43 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.e_br_ge_bqo_eiy |
| 29 | $C_2 \wr S_5$ | \( 1 + 4 T + 113 T^{2} + 416 T^{3} + 5930 T^{4} + 17208 T^{5} + 5930 p T^{6} + 416 p^{2} T^{7} + 113 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.e_ej_qa_iuc_zlw |
| 31 | $C_2 \wr S_5$ | \( 1 + 8 T + 125 T^{2} + 650 T^{3} + 6118 T^{4} + 24600 T^{5} + 6118 p T^{6} + 650 p^{2} T^{7} + 125 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.i_ev_za_jbi_bkke |
| 41 | $C_2 \wr S_5$ | \( 1 + 5 T + 197 T^{2} + 772 T^{3} + 15842 T^{4} + 46590 T^{5} + 15842 p T^{6} + 772 p^{2} T^{7} + 197 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.f_hp_bds_xli_cqxy |
| 43 | $C_2 \wr S_5$ | \( 1 + 10 T + 135 T^{2} + 968 T^{3} + 8506 T^{4} + 48796 T^{5} + 8506 p T^{6} + 968 p^{2} T^{7} + 135 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.k_ff_blg_mpe_cueu |
| 47 | $C_2 \wr S_5$ | \( 1 + 7 T + 143 T^{2} + 724 T^{3} + 8598 T^{4} + 38108 T^{5} + 8598 p T^{6} + 724 p^{2} T^{7} + 143 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.h_fn_bbw_mss_cejs |
| 53 | $C_2 \wr S_5$ | \( 1 + T + 121 T^{2} + 172 T^{3} + 10010 T^{4} + 13142 T^{5} + 10010 p T^{6} + 172 p^{2} T^{7} + 121 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.53.b_er_gq_ova_tlm |
| 59 | $C_2 \wr S_5$ | \( 1 - 30 T + 597 T^{2} - 8214 T^{3} + 89262 T^{4} - 759816 T^{5} + 89262 p T^{6} - 8214 p^{2} T^{7} + 597 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.abe_wz_amdy_fcbe_abrfzs |
| 61 | $C_2 \wr S_5$ | \( 1 + 14 T + 297 T^{2} + 2984 T^{3} + 35634 T^{4} + 263156 T^{5} + 35634 p T^{6} + 2984 p^{2} T^{7} + 297 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.o_ll_eku_caso_ozhk |
| 67 | $C_2 \wr S_5$ | \( 1 + 24 T + 429 T^{2} + 5118 T^{3} + 54298 T^{4} + 459388 T^{5} + 54298 p T^{6} + 5118 p^{2} T^{7} + 429 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.y_qn_how_dcik_badou |
| 71 | $C_2 \wr S_5$ | \( 1 - 7 T + 223 T^{2} - 356 T^{3} + 16278 T^{4} + 27126 T^{5} + 16278 p T^{6} - 356 p^{2} T^{7} + 223 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.ah_ip_ans_ycc_bodi |
| 73 | $C_2 \wr S_5$ | \( 1 - 5 T + 173 T^{2} - 1660 T^{3} + 13834 T^{4} - 188702 T^{5} + 13834 p T^{6} - 1660 p^{2} T^{7} + 173 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.af_gr_aclw_umc_aktdu |
| 79 | $C_2 \wr S_5$ | \( 1 + 28 T + 529 T^{2} + 7286 T^{3} + 85626 T^{4} + 821200 T^{5} + 85626 p T^{6} + 7286 p^{2} T^{7} + 529 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.bc_uj_kug_ewri_busuq |
| 83 | $C_2 \wr S_5$ | \( 1 + 27 T + 593 T^{2} + 8588 T^{3} + 108392 T^{4} + 1048784 T^{5} + 108392 p T^{6} + 8588 p^{2} T^{7} + 593 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.bb_wv_msi_geiy_chrlw |
| 89 | $C_2 \wr S_5$ | \( 1 - 6 T + 197 T^{2} - 1608 T^{3} + 29394 T^{4} - 168228 T^{5} + 29394 p T^{6} - 1608 p^{2} T^{7} + 197 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.ag_hp_acjw_brmo_ajowi |
| 97 | $C_2 \wr S_5$ | \( 1 + 26 T + 397 T^{2} + 2904 T^{3} + 4802 T^{4} - 92868 T^{5} + 4802 p T^{6} + 2904 p^{2} T^{7} + 397 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.ba_ph_ehs_hcs_afhjw |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.50161858650883499758728825204, −5.49672253798160849947206373929, −5.42119087663726254201007447650, −5.22061096574823682298447574406, −5.19233035991046395444126014430, −4.77389846871387830475818770869, −4.47293814114098997955554837725, −4.34386226779509351156400204615, −4.14573447902541874753868418282, −4.02325979997039230110701198783, −3.91118074396962997845364233890, −3.64901556236596374004241939179, −3.61709300233872081136384485912, −3.49416131160788962485651187865, −3.36780371711373686603880504702, −2.95541622176358851952492277817, −2.82286933037574744801654377514, −2.78435249631647625495597136897, −2.71949764687588940468497337979, −2.47141954323503575448142070348, −1.75169547983206026641032438849, −1.54035207492217391474610803279, −1.37286047707614436608066097285, −1.02909451604647836276138012946, −1.01719091867096492065771285146, 0, 0, 0, 0, 0,
1.01719091867096492065771285146, 1.02909451604647836276138012946, 1.37286047707614436608066097285, 1.54035207492217391474610803279, 1.75169547983206026641032438849, 2.47141954323503575448142070348, 2.71949764687588940468497337979, 2.78435249631647625495597136897, 2.82286933037574744801654377514, 2.95541622176358851952492277817, 3.36780371711373686603880504702, 3.49416131160788962485651187865, 3.61709300233872081136384485912, 3.64901556236596374004241939179, 3.91118074396962997845364233890, 4.02325979997039230110701198783, 4.14573447902541874753868418282, 4.34386226779509351156400204615, 4.47293814114098997955554837725, 4.77389846871387830475818770869, 5.19233035991046395444126014430, 5.22061096574823682298447574406, 5.42119087663726254201007447650, 5.49672253798160849947206373929, 5.50161858650883499758728825204
