| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 6·6-s − 11·7-s − 2·8-s − 5·11-s − 6·12-s − 4·13-s + 22·14-s + 16-s − 4·19-s + 33·21-s + 10·22-s − 4·23-s + 6·24-s + 8·26-s + 11·27-s − 22·28-s − 4·29-s + 8·31-s + 15·33-s − 5·37-s + 8·38-s + 12·39-s − 5·41-s − 66·42-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s − 4.15·7-s − 0.707·8-s − 1.50·11-s − 1.73·12-s − 1.10·13-s + 5.87·14-s + 1/4·16-s − 0.917·19-s + 7.20·21-s + 2.13·22-s − 0.834·23-s + 1.22·24-s + 1.56·26-s + 2.11·27-s − 4.15·28-s − 0.742·29-s + 1.43·31-s + 2.61·33-s − 0.821·37-s + 1.29·38-s + 1.92·39-s − 0.780·41-s − 10.1·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) | |
| 37 | $C_1$ | \( ( 1 + T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + p T + p T^{2} + p T^{3} + 3 T^{4} + p^{2} T^{5} + 3 p T^{6} + p^{3} T^{7} + p^{4} T^{8} + p^{5} T^{9} + p^{5} T^{10} \) | 5.2.c_c_c_d_e |
| 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.f_bv_gq_bla_dys |
| 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.e_bl_fs_bcs_ebw |
| 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_am_ww_axc |
| 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.e_cf_je_cgo_jcm |
| 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.ai_ev_aza_jbi_abkke |
| 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.ab_er_agq_ova_atlm |
| 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.be_wz_mdy_fcbe_brfzs |
| 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.h_ip_ns_ycc_abodi |
| 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.f_gr_clw_umc_ktdu |
| 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.abc_uj_akug_ewri_abusuq |
| 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.aba_ph_aehs_hcs_fhjw |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.45487462924071795403071494059, −6.35516781159197621411732743103, −6.30257167866244476214083906112, −6.10724984175265552899618452804, −6.10209691831214886062966583163, −5.78727034243573425325669843597, −5.40184522088152370175664060989, −5.38991034642070123500025090993, −5.26015093325283557640151474221, −5.10996007972715214353682137927, −4.54091985411144372914199966036, −4.52858117071984156424919149680, −4.37713892747275704833236154306, −4.22586525859944265601599559748, −3.50705389242212901516020800369, −3.42515862379680679266669148520, −3.35621078622605056764104711943, −3.18510171781422560197161661049, −2.92477936020162100598330330671, −2.67472831736338453149198563234, −2.49761115346100632138825496932, −2.40887773468964179732949233343, −1.81501702475022691690717919881, −1.40876692429763872414559496937, −1.21248584631441819193752730707, 0, 0, 0, 0, 0,
1.21248584631441819193752730707, 1.40876692429763872414559496937, 1.81501702475022691690717919881, 2.40887773468964179732949233343, 2.49761115346100632138825496932, 2.67472831736338453149198563234, 2.92477936020162100598330330671, 3.18510171781422560197161661049, 3.35621078622605056764104711943, 3.42515862379680679266669148520, 3.50705389242212901516020800369, 4.22586525859944265601599559748, 4.37713892747275704833236154306, 4.52858117071984156424919149680, 4.54091985411144372914199966036, 5.10996007972715214353682137927, 5.26015093325283557640151474221, 5.38991034642070123500025090993, 5.40184522088152370175664060989, 5.78727034243573425325669843597, 6.10209691831214886062966583163, 6.10724984175265552899618452804, 6.30257167866244476214083906112, 6.35516781159197621411732743103, 6.45487462924071795403071494059
