| L(s) = 1 | − 2-s − 5·3-s − 4-s − 4·5-s + 5·6-s + 5·7-s + 5·8-s + 15·9-s + 4·10-s + 5·12-s + 3·13-s − 5·14-s + 20·15-s − 5·16-s − 15·18-s − 19-s + 4·20-s − 25·21-s − 4·23-s − 25·24-s + 6·25-s − 3·26-s − 35·27-s − 5·28-s − 13·29-s − 20·30-s − 13·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.88·3-s − 1/2·4-s − 1.78·5-s + 2.04·6-s + 1.88·7-s + 1.76·8-s + 5·9-s + 1.26·10-s + 1.44·12-s + 0.832·13-s − 1.33·14-s + 5.16·15-s − 5/4·16-s − 3.53·18-s − 0.229·19-s + 0.894·20-s − 5.45·21-s − 0.834·23-s − 5.10·24-s + 6/5·25-s − 0.588·26-s − 6.73·27-s − 0.944·28-s − 2.41·29-s − 3.65·30-s − 2.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{5} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{5} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 7 | $C_1$ | \( ( 1 - T )^{5} \) | |
| 17 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + T + p T^{2} - p T^{3} - 7 T^{5} - p^{3} T^{7} + p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.2.b_c_ac_a_ah |
| 5 | $C_2 \wr S_5$ | \( 1 + 4 T + 2 p T^{2} + 7 T^{3} - 7 T^{4} - 66 T^{5} - 7 p T^{6} + 7 p^{2} T^{7} + 2 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.5.e_k_h_ah_aco |
| 11 | $C_2 \wr S_5$ | \( 1 + 2 p T^{2} + 25 T^{3} + 241 T^{4} + 614 T^{5} + 241 p T^{6} + 25 p^{2} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \) | 5.11.a_w_z_jh_xq |
| 13 | $C_2 \wr S_5$ | \( 1 - 3 T + 51 T^{2} - 123 T^{3} + 1198 T^{4} - 2252 T^{5} + 1198 p T^{6} - 123 p^{2} T^{7} + 51 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.ad_bz_aet_buc_adiq |
| 19 | $C_2 \wr S_5$ | \( 1 + T + 43 T^{2} + 17 T^{3} + 882 T^{4} - 200 T^{5} + 882 p T^{6} + 17 p^{2} T^{7} + 43 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.19.b_br_r_bhy_ahs |
| 23 | $C_2 \wr S_5$ | \( 1 + 4 T + 43 T^{2} + 5 p T^{3} + 1439 T^{4} + 4597 T^{5} + 1439 p T^{6} + 5 p^{3} T^{7} + 43 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.e_br_el_cdj_guv |
| 29 | $C_2 \wr S_5$ | \( 1 + 13 T + 179 T^{2} + 1462 T^{3} + 11301 T^{4} + 62963 T^{5} + 11301 p T^{6} + 1462 p^{2} T^{7} + 179 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.n_gx_ceg_qsr_dpdr |
| 31 | $C_2 \wr S_5$ | \( 1 + 13 T + 4 p T^{2} + 843 T^{3} + 6245 T^{4} + 36412 T^{5} + 6245 p T^{6} + 843 p^{2} T^{7} + 4 p^{4} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.n_eu_bgl_jgf_cbwm |
| 37 | $C_2 \wr S_5$ | \( 1 + 11 T + 206 T^{2} + 1612 T^{3} + 15973 T^{4} + 89153 T^{5} + 15973 p T^{6} + 1612 p^{2} T^{7} + 206 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.37.l_hy_cka_xqj_fbwz |
| 41 | $C_2 \wr S_5$ | \( 1 - T + 113 T^{2} + 75 T^{3} + 6090 T^{4} + 9636 T^{5} + 6090 p T^{6} + 75 p^{2} T^{7} + 113 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.41.ab_ej_cx_jag_ogq |
| 43 | $C_2 \wr S_5$ | \( 1 - T + 30 T^{2} - 266 T^{3} + 1669 T^{4} - 5107 T^{5} + 1669 p T^{6} - 266 p^{2} T^{7} + 30 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.43.ab_be_akg_cmf_ahol |
| 47 | $C_2 \wr S_5$ | \( 1 + 4 T + 210 T^{2} + 589 T^{3} + 18275 T^{4} + 37538 T^{5} + 18275 p T^{6} + 589 p^{2} T^{7} + 210 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.e_ic_wr_bbax_cdnu |
| 53 | $C_2 \wr S_5$ | \( 1 + 19 T + 378 T^{2} + 4126 T^{3} + 44951 T^{4} + 328013 T^{5} + 44951 p T^{6} + 4126 p^{2} T^{7} + 378 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.t_oo_gcs_comx_srfx |
| 59 | $C_2 \wr S_5$ | \( 1 - 3 T + 141 T^{2} - 41 T^{3} + 9630 T^{4} + 7820 T^{5} + 9630 p T^{6} - 41 p^{2} T^{7} + 141 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.ad_fl_abp_ogk_lou |
| 61 | $C_2 \wr S_5$ | \( 1 - 5 T + 129 T^{2} - 629 T^{3} + 12458 T^{4} - 47364 T^{5} + 12458 p T^{6} - 629 p^{2} T^{7} + 129 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.af_ez_ayf_sle_acsbs |
| 67 | $C_2 \wr S_5$ | \( 1 + 11 T + 234 T^{2} + 2225 T^{3} + 26993 T^{4} + 210384 T^{5} + 26993 p T^{6} + 2225 p^{2} T^{7} + 234 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.l_ja_dhp_bnyf_lzfs |
| 71 | $C_2 \wr S_5$ | \( 1 + 31 T + 677 T^{2} + 10002 T^{3} + 119521 T^{4} + 1105953 T^{5} + 119521 p T^{6} + 10002 p^{2} T^{7} + 677 p^{3} T^{8} + 31 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.bf_bab_ous_guuz_ckyar |
| 73 | $C_2 \wr S_5$ | \( 1 + 13 T + 222 T^{2} + 1435 T^{3} + 15587 T^{4} + 77520 T^{5} + 15587 p T^{6} + 1435 p^{2} T^{7} + 222 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.n_io_cdf_xbn_ekro |
| 79 | $C_2 \wr S_5$ | \( 1 + 5 T + 273 T^{2} + 866 T^{3} + 33857 T^{4} + 79239 T^{5} + 33857 p T^{6} + 866 p^{2} T^{7} + 273 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.f_kn_bhi_bycf_enfr |
| 83 | $C_2 \wr S_5$ | \( 1 + 2 T + 122 T^{2} - 377 T^{3} + 17717 T^{4} + 17826 T^{5} + 17717 p T^{6} - 377 p^{2} T^{7} + 122 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.c_es_aon_bafl_bajq |
| 89 | $C_2 \wr S_5$ | \( 1 - 5 T + 270 T^{2} - 839 T^{3} + 33831 T^{4} - 76528 T^{5} + 33831 p T^{6} - 839 p^{2} T^{7} + 270 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.af_kk_abgh_bybf_aejfk |
| 97 | $C_2 \wr S_5$ | \( 1 - 3 T + 73 T^{2} + 997 T^{3} + 17676 T^{4} - 58860 T^{5} + 17676 p T^{6} + 997 p^{2} T^{7} + 73 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.ad_cv_bmj_badw_adjbw |
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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
−5.12463147550026326268946605209, −4.88385645861917764766462175816, −4.78745887961738918717926609746, −4.78128858246116632579386710787, −4.76040378605725768603538610737, −4.26204883509625389606957284718, −4.23590251431363178833717341810, −4.04312158611968137395065474569, −4.02773466977798137618096737947, −3.97396636102322370311772659864, −3.66966867753732391769012948894, −3.65497147997096026848603404571, −3.32727377179594895358482666826, −3.25993603424249103766529713367, −2.75145946335149272902437001750, −2.66560915153251759170309286602, −2.51964216312731491362794533503, −1.92978997529212502110553404184, −1.88918095103163547700151800563, −1.77447560427482768153402674746, −1.70023337633236284852721302245, −1.31628902308745328328553720063, −1.21741262779211422368335122093, −1.13694455732998335274190983076, −0.958759670122094566349840309978, 0, 0, 0, 0, 0,
0.958759670122094566349840309978, 1.13694455732998335274190983076, 1.21741262779211422368335122093, 1.31628902308745328328553720063, 1.70023337633236284852721302245, 1.77447560427482768153402674746, 1.88918095103163547700151800563, 1.92978997529212502110553404184, 2.51964216312731491362794533503, 2.66560915153251759170309286602, 2.75145946335149272902437001750, 3.25993603424249103766529713367, 3.32727377179594895358482666826, 3.65497147997096026848603404571, 3.66966867753732391769012948894, 3.97396636102322370311772659864, 4.02773466977798137618096737947, 4.04312158611968137395065474569, 4.23590251431363178833717341810, 4.26204883509625389606957284718, 4.76040378605725768603538610737, 4.78128858246116632579386710787, 4.78745887961738918717926609746, 4.88385645861917764766462175816, 5.12463147550026326268946605209
