| L(s) = 1 | − 4·3-s − 3·5-s − 4·7-s + 10·9-s + 2·11-s + 4·13-s + 12·15-s − 2·17-s − 7·19-s + 16·21-s − 3·23-s − 25-s − 20·27-s + 29-s − 3·31-s − 8·33-s + 12·35-s + 10·37-s − 16·39-s − 16·41-s − 3·43-s − 30·45-s − 5·47-s + 10·49-s + 8·51-s + 5·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.34·5-s − 1.51·7-s + 10/3·9-s + 0.603·11-s + 1.10·13-s + 3.09·15-s − 0.485·17-s − 1.60·19-s + 3.49·21-s − 0.625·23-s − 1/5·25-s − 3.84·27-s + 0.185·29-s − 0.538·31-s − 1.39·33-s + 2.02·35-s + 1.64·37-s − 2.56·39-s − 2.49·41-s − 0.457·43-s − 4.47·45-s − 0.729·47-s + 10/7·49-s + 1.12·51-s + 0.686·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 3 T + 2 p T^{2} + p^{2} T^{3} + 74 T^{4} + p^{3} T^{5} + 2 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.d_k_z_cw |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 20 T^{2} - 34 T^{3} + 294 T^{4} - 34 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_u_abi_li |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 40 T^{2} + 62 T^{3} + 878 T^{4} + 62 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.c_bo_ck_bhu |
| 19 | $C_2 \wr S_4$ | \( 1 + 7 T + 64 T^{2} + 351 T^{3} + 1774 T^{4} + 351 p T^{5} + 64 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.h_cm_nn_cqg |
| 23 | $C_2 \wr S_4$ | \( 1 + 3 T + 40 T^{2} - 49 T^{3} + 494 T^{4} - 49 p T^{5} + 40 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.d_bo_abx_ta |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 86 T^{2} - 35 T^{3} + 3378 T^{4} - 35 p T^{5} + 86 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ab_di_abj_ezy |
| 31 | $C_2 \wr S_4$ | \( 1 + 3 T - 4 T^{2} + 119 T^{3} + 58 p T^{4} + 119 p T^{5} - 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.d_ae_ep_cre |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 270 T^{3} + 1870 T^{4} - 270 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ak_cm_akk_cty |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 4 p T^{2} + 1280 T^{3} + 8694 T^{4} + 1280 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.q_gi_bxg_mwk |
| 43 | $C_2 \wr S_4$ | \( 1 + 3 T + 128 T^{2} + 275 T^{3} + 7246 T^{4} + 275 p T^{5} + 128 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.d_ey_kp_kss |
| 47 | $C_2 \wr S_4$ | \( 1 + 5 T + 148 T^{2} + 689 T^{3} + 9638 T^{4} + 689 p T^{5} + 148 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.f_fs_ban_ogs |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 174 T^{2} - 727 T^{3} + 12802 T^{4} - 727 p T^{5} + 174 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.af_gs_abbz_syk |
| 59 | $C_2 \wr S_4$ | \( 1 - 20 T + 316 T^{2} - 3236 T^{3} + 28790 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.au_me_aeum_bqpi |
| 61 | $C_2 \wr S_4$ | \( 1 - 12 T + 180 T^{2} - 1508 T^{3} + 15014 T^{4} - 1508 p T^{5} + 180 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.am_gy_acga_wfm |
| 67 | $C_2 \wr S_4$ | \( 1 - 22 T + 228 T^{2} - 1254 T^{3} + 6086 T^{4} - 1254 p T^{5} + 228 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.aw_iu_abwg_jac |
| 71 | $C_2 \wr S_4$ | \( 1 + 52 T^{2} - 304 T^{3} + 7478 T^{4} - 304 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ca_als_lbq |
| 73 | $C_2 \wr S_4$ | \( 1 + 13 T + 126 T^{2} - 261 T^{3} - 3934 T^{4} - 261 p T^{5} + 126 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.n_ew_akb_afvi |
| 79 | $C_2 \wr S_4$ | \( 1 + 11 T + 196 T^{2} + 1167 T^{3} + 15030 T^{4} + 1167 p T^{5} + 196 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.l_ho_bsx_wgc |
| 83 | $C_2 \wr S_4$ | \( 1 + T + 296 T^{2} + 329 T^{3} + 35310 T^{4} + 329 p T^{5} + 296 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.83.b_lk_mr_cagc |
| 89 | $C_2 \wr S_4$ | \( 1 + 5 T + 194 T^{2} + 139 T^{3} + 16986 T^{4} + 139 p T^{5} + 194 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.f_hm_fj_zdi |
| 97 | $C_2 \wr S_4$ | \( 1 + 17 T + 374 T^{2} + 4127 T^{3} + 52210 T^{4} + 4127 p T^{5} + 374 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.r_ok_gct_czgc |
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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
−6.53434125121001600912769158801, −6.01828820532158172226743560416, −5.78638959430261682296768181874, −5.74003334418831623223860166038, −5.69706293111898123488290133480, −5.37770946520709482754223670889, −5.11452076224163019924266588334, −5.06541436665969022111322355608, −4.75927787377057189179851769076, −4.46295398681368225132995224340, −4.13896765872572997948131892815, −4.05272872141826898148252331157, −4.04804755671374475120997536793, −3.81646177768785286233772553466, −3.62096671539881266056804701561, −3.54612224333131570226586081691, −3.21462714442190887873889488905, −2.55137885659551782993702748081, −2.52541767103841029529796593057, −2.51087080987043957889307459634, −2.11990469519213814216392058765, −1.57616668352890074599738179797, −1.29374911486001396090487841606, −1.09618222228326484542116341695, −1.06388457822202578151231961597, 0, 0, 0, 0,
1.06388457822202578151231961597, 1.09618222228326484542116341695, 1.29374911486001396090487841606, 1.57616668352890074599738179797, 2.11990469519213814216392058765, 2.51087080987043957889307459634, 2.52541767103841029529796593057, 2.55137885659551782993702748081, 3.21462714442190887873889488905, 3.54612224333131570226586081691, 3.62096671539881266056804701561, 3.81646177768785286233772553466, 4.04804755671374475120997536793, 4.05272872141826898148252331157, 4.13896765872572997948131892815, 4.46295398681368225132995224340, 4.75927787377057189179851769076, 5.06541436665969022111322355608, 5.11452076224163019924266588334, 5.37770946520709482754223670889, 5.69706293111898123488290133480, 5.74003334418831623223860166038, 5.78638959430261682296768181874, 6.01828820532158172226743560416, 6.53434125121001600912769158801
