| L(s) = 1 | − 2-s + 5·3-s − 3·4-s − 2·5-s − 5·6-s − 6·7-s + 8-s + 15·9-s + 2·10-s − 15·12-s + 5·13-s + 6·14-s − 10·15-s + 8·16-s − 2·17-s − 15·18-s − 12·19-s + 6·20-s − 30·21-s + 8·23-s + 5·24-s − 7·25-s − 5·26-s + 35·27-s + 18·28-s − 6·29-s + 10·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.88·3-s − 3/2·4-s − 0.894·5-s − 2.04·6-s − 2.26·7-s + 0.353·8-s + 5·9-s + 0.632·10-s − 4.33·12-s + 1.38·13-s + 1.60·14-s − 2.58·15-s + 2·16-s − 0.485·17-s − 3.53·18-s − 2.75·19-s + 1.34·20-s − 6.54·21-s + 1.66·23-s + 1.02·24-s − 7/5·25-s − 0.980·26-s + 6.73·27-s + 3.40·28-s − 1.11·29-s + 1.82·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 11^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 11^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{5} \) | |
| 11 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + T + p^{2} T^{2} + 3 p T^{3} + 9 T^{4} + 15 T^{5} + 9 p T^{6} + 3 p^{3} T^{7} + p^{5} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.2.b_e_g_j_p |
| 5 | $C_2 \wr S_5$ | \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 62 T^{4} + 62 T^{5} + 62 p T^{6} + 16 p^{2} T^{7} + 11 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.5.c_l_q_ck_ck |
| 7 | $C_2 \wr S_5$ | \( 1 + 6 T + 41 T^{2} + 156 T^{3} + 600 T^{4} + 1592 T^{5} + 600 p T^{6} + 156 p^{2} T^{7} + 41 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.g_bp_ga_xc_cjg |
| 17 | $C_2 \wr S_5$ | \( 1 + 2 T + 47 T^{2} + 234 T^{3} + 864 T^{4} + 6826 T^{5} + 864 p T^{6} + 234 p^{2} T^{7} + 47 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.17.c_bv_ja_bhg_kco |
| 19 | $C_2 \wr S_5$ | \( 1 + 12 T + 105 T^{2} + 564 T^{3} + 2768 T^{4} + 11204 T^{5} + 2768 p T^{6} + 564 p^{2} T^{7} + 105 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.m_eb_vs_ecm_qoy |
| 23 | $C_2 \wr S_5$ | \( 1 - 8 T + 79 T^{2} - 400 T^{3} + 2746 T^{4} - 11452 T^{5} + 2746 p T^{6} - 400 p^{2} T^{7} + 79 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.ai_db_apk_ebq_aqym |
| 29 | $C_2 \wr S_5$ | \( 1 + 6 T + 59 T^{2} + 248 T^{3} + 1380 T^{4} + 5490 T^{5} + 1380 p T^{6} + 248 p^{2} T^{7} + 59 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.g_ch_jo_cbc_ide |
| 31 | $C_2 \wr S_5$ | \( 1 + 4 T + 37 T^{2} + 128 T^{3} + 26 p T^{4} - 1614 T^{5} + 26 p^{2} T^{6} + 128 p^{2} T^{7} + 37 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.e_bl_ey_bfa_ackc |
| 37 | $C_2 \wr S_5$ | \( 1 + 10 T + 105 T^{2} + 532 T^{3} + 3506 T^{4} + 12172 T^{5} + 3506 p T^{6} + 532 p^{2} T^{7} + 105 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.k_eb_um_few_sae |
| 41 | $C_2 \wr S_5$ | \( 1 + 6 T + 3 p T^{2} + 404 T^{3} + 6984 T^{4} + 17592 T^{5} + 6984 p T^{6} + 404 p^{2} T^{7} + 3 p^{4} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.g_et_po_kiq_baaq |
| 43 | $C_2 \wr S_5$ | \( 1 + 26 T + 457 T^{2} + 5438 T^{3} + 51160 T^{4} + 371774 T^{5} + 51160 p T^{6} + 5438 p^{2} T^{7} + 457 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.ba_rp_ibe_cxrs_vdza |
| 47 | $C_2 \wr S_5$ | \( 1 + 16 T + 203 T^{2} + 1596 T^{3} + 10986 T^{4} + 70352 T^{5} + 10986 p T^{6} + 1596 p^{2} T^{7} + 203 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.q_hv_cjk_qgo_eabw |
| 53 | $C_2 \wr S_5$ | \( 1 - 6 T + 181 T^{2} - 1280 T^{3} + 15326 T^{4} - 101988 T^{5} + 15326 p T^{6} - 1280 p^{2} T^{7} + 181 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.ag_gz_abxg_wrm_afuwq |
| 59 | $C_2 \wr S_5$ | \( 1 - 6 T + 119 T^{2} - 424 T^{3} + 10938 T^{4} - 47460 T^{5} + 10938 p T^{6} - 424 p^{2} T^{7} + 119 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.ag_ep_aqi_qes_acsfk |
| 61 | $C_2 \wr S_5$ | \( 1 + 14 T + 205 T^{2} + 1688 T^{3} + 19054 T^{4} + 135884 T^{5} + 19054 p T^{6} + 1688 p^{2} T^{7} + 205 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.o_hx_cmy_bcew_htai |
| 67 | $C_2 \wr S_5$ | \( 1 + 2 T + 321 T^{2} + 512 T^{3} + 42098 T^{4} + 50654 T^{5} + 42098 p T^{6} + 512 p^{2} T^{7} + 321 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.c_mj_ts_ckhe_cwyg |
| 71 | $C_2 \wr S_5$ | \( 1 + 14 T + 363 T^{2} + 3676 T^{3} + 51282 T^{4} + 380356 T^{5} + 51282 p T^{6} + 3676 p^{2} T^{7} + 363 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.o_nz_flk_cxwk_vqrc |
| 73 | $C_2 \wr S_5$ | \( 1 + 12 T + 211 T^{2} + 1552 T^{3} + 24196 T^{4} + 166548 T^{5} + 24196 p T^{6} + 1552 p^{2} T^{7} + 211 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.m_id_chs_bjuq_jmjs |
| 79 | $C_2 \wr S_5$ | \( 1 - 8 T + 213 T^{2} - 1470 T^{3} + 28148 T^{4} - 158970 T^{5} + 28148 p T^{6} - 1470 p^{2} T^{7} + 213 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.ai_if_aceo_bpqq_ajbeg |
| 83 | $C_2 \wr S_5$ | \( 1 - 22 T + 475 T^{2} - 6072 T^{3} + 75990 T^{4} - 694068 T^{5} + 75990 p T^{6} - 6072 p^{2} T^{7} + 475 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.aw_sh_aizo_eiks_abnmsy |
| 89 | $C_2 \wr S_5$ | \( 1 + 8 T + 247 T^{2} + 1410 T^{3} + 28066 T^{4} + 125518 T^{5} + 28066 p T^{6} + 1410 p^{2} T^{7} + 247 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.i_jn_ccg_bpnm_hdrq |
| 97 | $C_2 \wr S_5$ | \( 1 + 121 T^{2} + 524 T^{3} + 19814 T^{4} + 30128 T^{5} + 19814 p T^{6} + 524 p^{2} T^{7} + 121 p^{3} T^{8} + p^{5} T^{10} \) | 5.97.a_er_ue_bdic_bsou |
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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
−5.16749971548900131225417551007, −5.04226901621193533084849373408, −4.84307428917368886349893495819, −4.80401247096659925986649047763, −4.62494772470779718723200472982, −4.19948973143988267695183182079, −4.18548410564031998780256751413, −4.13606770431498081966260060260, −3.94014970066950473109447187521, −3.71152148774509271077486159982, −3.54251743117695959807259547517, −3.44008413562501002155015423271, −3.33638237728358940352070008373, −3.27971801523922362322396669694, −3.01319789790444044027255907744, −3.00639877877724418414059120966, −2.89262563027117219091095331198, −2.31552984815332449755800962846, −2.17241311093693570825776146005, −2.16093481600583158513359523142, −1.78712858031013261269951092920, −1.72317133095236701019869727247, −1.37145763608728269599110646077, −1.20148914674242212320078319625, −1.10159249790340562330130830019, 0, 0, 0, 0, 0,
1.10159249790340562330130830019, 1.20148914674242212320078319625, 1.37145763608728269599110646077, 1.72317133095236701019869727247, 1.78712858031013261269951092920, 2.16093481600583158513359523142, 2.17241311093693570825776146005, 2.31552984815332449755800962846, 2.89262563027117219091095331198, 3.00639877877724418414059120966, 3.01319789790444044027255907744, 3.27971801523922362322396669694, 3.33638237728358940352070008373, 3.44008413562501002155015423271, 3.54251743117695959807259547517, 3.71152148774509271077486159982, 3.94014970066950473109447187521, 4.13606770431498081966260060260, 4.18548410564031998780256751413, 4.19948973143988267695183182079, 4.62494772470779718723200472982, 4.80401247096659925986649047763, 4.84307428917368886349893495819, 5.04226901621193533084849373408, 5.16749971548900131225417551007
