| L(s) = 1 | + 4·2-s + 10·4-s − 4·5-s − 4·7-s + 20·8-s − 2·9-s − 16·10-s + 3·11-s − 8·13-s − 16·14-s + 35·16-s + 5·17-s − 8·18-s − 3·19-s − 40·20-s + 12·22-s − 32·26-s + 27-s − 40·28-s + 9·29-s − 4·31-s + 56·32-s + 20·34-s + 16·35-s − 20·36-s − 4·37-s − 12·38-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 1.78·5-s − 1.51·7-s + 7.07·8-s − 2/3·9-s − 5.05·10-s + 0.904·11-s − 2.21·13-s − 4.27·14-s + 35/4·16-s + 1.21·17-s − 1.88·18-s − 0.688·19-s − 8.94·20-s + 2.55·22-s − 6.27·26-s + 0.192·27-s − 7.55·28-s + 1.67·29-s − 0.718·31-s + 9.89·32-s + 3.42·34-s + 2.70·35-s − 3.33·36-s − 0.657·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 2 T^{2} - T^{3} - 2 T^{4} - p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_c_ab_ac |
| 5 | $C_2 \wr S_4$ | \( 1 + 4 T + 16 T^{2} + 9 p T^{3} + 106 T^{4} + 9 p^{2} T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.e_q_bt_ec |
| 11 | $C_2 \wr S_4$ | \( 1 - 3 T + 13 T^{2} - 31 T^{3} + 108 T^{4} - 31 p T^{5} + 13 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ad_n_abf_ee |
| 13 | $C_2 \wr S_4$ | \( 1 + 8 T + 34 T^{2} + 177 T^{3} + 64 p T^{4} + 177 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.i_bi_gv_bga |
| 17 | $C_2 \wr S_4$ | \( 1 - 5 T + 29 T^{2} - 142 T^{3} + 906 T^{4} - 142 p T^{5} + 29 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.af_bd_afm_biw |
| 19 | $C_2 \wr S_4$ | \( 1 + 3 T + 39 T^{2} + 92 T^{3} + 784 T^{4} + 92 p T^{5} + 39 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.d_bn_do_bee |
| 29 | $C_2 \wr S_4$ | \( 1 - 9 T + 98 T^{2} - 611 T^{3} + 4354 T^{4} - 611 p T^{5} + 98 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.aj_du_axn_glm |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 81 T^{2} + 342 T^{3} + 3372 T^{4} + 342 p T^{5} + 81 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.e_dd_ne_ezs |
| 37 | $C_2 \wr S_4$ | \( 1 + 4 T + 105 T^{2} + 286 T^{3} + 5176 T^{4} + 286 p T^{5} + 105 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_eb_la_hrc |
| 41 | $C_2 \wr S_4$ | \( 1 + 88 T^{2} - 289 T^{3} + 3596 T^{4} - 289 p T^{5} + 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_dk_ald_fii |
| 43 | $C_2 \wr S_4$ | \( 1 + 13 T + 123 T^{2} + 1084 T^{3} + 8956 T^{4} + 1084 p T^{5} + 123 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.n_et_bps_ngm |
| 47 | $C_2 \wr S_4$ | \( 1 + T + 111 T^{2} - 199 T^{3} + 5632 T^{4} - 199 p T^{5} + 111 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.47.b_eh_ahr_iiq |
| 53 | $C_2 \wr S_4$ | \( 1 + 17 T + 272 T^{2} + 2599 T^{3} + 23238 T^{4} + 2599 p T^{5} + 272 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.r_km_dvz_biju |
| 59 | $C_2 \wr S_4$ | \( 1 + 7 T + 170 T^{2} + 699 T^{3} + 12234 T^{4} + 699 p T^{5} + 170 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.h_go_bax_sco |
| 61 | $C_2 \wr S_4$ | \( 1 + 33 T + 604 T^{2} + 7487 T^{3} + 68158 T^{4} + 7487 p T^{5} + 604 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.bh_xg_lbz_dwvm |
| 67 | $C_2 \wr S_4$ | \( 1 + T + 200 T^{2} - 71 T^{3} + 17566 T^{4} - 71 p T^{5} + 200 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.67.b_hs_act_zzq |
| 71 | $C_2 \wr S_4$ | \( 1 + 35 T + 677 T^{2} + 8961 T^{3} + 87628 T^{4} + 8961 p T^{5} + 677 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.bj_bab_ngr_ezqi |
| 73 | $C_2 \wr S_4$ | \( 1 + 10 T + 168 T^{2} + 1284 T^{3} + 13481 T^{4} + 1284 p T^{5} + 168 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.k_gm_bxk_tyn |
| 79 | $C_2 \wr S_4$ | \( 1 + 12 T + 247 T^{2} + 1544 T^{3} + 22576 T^{4} + 1544 p T^{5} + 247 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.m_jn_chk_bhki |
| 83 | $C_2 \wr S_4$ | \( 1 + 12 T + 344 T^{2} + 2843 T^{3} + 43462 T^{4} + 2843 p T^{5} + 344 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.m_ng_efj_cmhq |
| 89 | $C_2 \wr S_4$ | \( 1 + 3 T + 253 T^{2} + 850 T^{3} + 29378 T^{4} + 850 p T^{5} + 253 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.d_jt_bgs_brly |
| 97 | $C_2 \wr S_4$ | \( 1 - 5 T + 9 T^{2} + 258 T^{3} + 8406 T^{4} + 258 p T^{5} + 9 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.af_j_jy_mli |
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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
−5.89546290397040831935135083928, −5.75378675924379560905007829547, −5.65411540849899920634489498322, −5.33682789043002034022577949282, −4.91710541375815932031860087147, −4.87833991152849174804939255443, −4.78991377693115944107158170809, −4.46533939104497374468321653589, −4.44039965450253287854013391329, −4.42174429948420238833500697640, −4.10922107784146585949405230467, −3.80560780480601516450492076926, −3.64891240876972491041663496855, −3.33269185624431730789764832531, −3.26815446715331075518538057678, −3.19333758495874698209206759543, −3.13641913677285138658293160700, −2.74227088707360559890765159381, −2.63753062490435831337424537321, −2.34041912788349727359335580940, −2.24354529104947159801449193325, −1.69605703793822079265150285187, −1.38868748578358725909671745812, −1.36513365509348545548053913345, −1.22608784023212553541723377258, 0, 0, 0, 0,
1.22608784023212553541723377258, 1.36513365509348545548053913345, 1.38868748578358725909671745812, 1.69605703793822079265150285187, 2.24354529104947159801449193325, 2.34041912788349727359335580940, 2.63753062490435831337424537321, 2.74227088707360559890765159381, 3.13641913677285138658293160700, 3.19333758495874698209206759543, 3.26815446715331075518538057678, 3.33269185624431730789764832531, 3.64891240876972491041663496855, 3.80560780480601516450492076926, 4.10922107784146585949405230467, 4.42174429948420238833500697640, 4.44039965450253287854013391329, 4.46533939104497374468321653589, 4.78991377693115944107158170809, 4.87833991152849174804939255443, 4.91710541375815932031860087147, 5.33682789043002034022577949282, 5.65411540849899920634489498322, 5.75378675924379560905007829547, 5.89546290397040831935135083928
