| L(s) = 1 | + 2-s − 3-s − 4-s − 7·5-s − 6-s − 2·9-s − 7·10-s + 11-s + 12-s + 7·15-s + 4·16-s + 4·17-s − 2·18-s + 19-s + 7·20-s + 22-s − 2·23-s + 17·25-s + 12·27-s + 29-s + 7·30-s − 4·31-s + 6·32-s − 33-s + 4·34-s + 2·36-s + 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 3.13·5-s − 0.408·6-s − 2/3·9-s − 2.21·10-s + 0.301·11-s + 0.288·12-s + 1.80·15-s + 16-s + 0.970·17-s − 0.471·18-s + 0.229·19-s + 1.56·20-s + 0.213·22-s − 0.417·23-s + 17/5·25-s + 2.30·27-s + 0.185·29-s + 1.27·30-s − 0.718·31-s + 1.06·32-s − 0.174·33-s + 0.685·34-s + 1/3·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7680981413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7680981413\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | | \( 1 \) | |
| 13 | | \( 1 \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + p T^{2} - 3 T^{3} + T^{4} - 3 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.2.ab_c_ad_b |
| 3 | $C_2 \wr S_4$ | \( 1 + T + p T^{2} - 7 T^{3} - 2 p T^{4} - 7 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.3.b_d_ah_ag |
| 5 | $C_2 \wr S_4$ | \( 1 + 7 T + 32 T^{2} + 104 T^{3} + 261 T^{4} + 104 p T^{5} + 32 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.h_bg_ea_kb |
| 11 | $C_2 \wr S_4$ | \( 1 - T + 35 T^{2} - 17 T^{3} + 522 T^{4} - 17 p T^{5} + 35 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ab_bj_ar_uc |
| 17 | $C_2 \wr S_4$ | \( 1 - 4 T + 56 T^{2} - 144 T^{3} + 1273 T^{4} - 144 p T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_ce_afo_bwz |
| 19 | $C_2 \wr S_4$ | \( 1 - T + 21 T^{2} - 3 p T^{3} + 576 T^{4} - 3 p^{2} T^{5} + 21 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ab_v_acf_we |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 54 T^{2} + 82 T^{3} + 1498 T^{4} + 82 p T^{5} + 54 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.c_cc_de_cfq |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 94 T^{2} - 108 T^{3} + 3765 T^{4} - 108 p T^{5} + 94 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ab_dq_aee_fov |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 111 T^{2} + 346 T^{3} + 5014 T^{4} + 346 p T^{5} + 111 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.e_eh_ni_hkw |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 165 T^{2} - 1066 T^{3} + 9456 T^{4} - 1066 p T^{5} + 165 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ak_gj_abpa_nzs |
| 41 | $C_2 \wr S_4$ | \( 1 + 22 T + 313 T^{2} + 2932 T^{3} + 21740 T^{4} + 2932 p T^{5} + 313 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.w_mb_eiu_bgee |
| 43 | $C_2 \wr S_4$ | \( 1 - 3 T + 169 T^{2} - 379 T^{3} + 10834 T^{4} - 379 p T^{5} + 169 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ad_gn_aop_qas |
| 47 | $C_2 \wr S_4$ | \( 1 - 2 T + 137 T^{2} - 294 T^{3} + 8560 T^{4} - 294 p T^{5} + 137 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ac_fh_ali_mrg |
| 53 | $C_2 \wr S_4$ | \( 1 - 2 T + 72 T^{2} + 572 T^{3} + 625 T^{4} + 572 p T^{5} + 72 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ac_cu_wa_yb |
| 59 | $C_2 \wr S_4$ | \( 1 + 8 T + 159 T^{2} + 866 T^{3} + 11094 T^{4} + 866 p T^{5} + 159 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.i_gd_bhi_qks |
| 61 | $C_2 \wr S_4$ | \( 1 + 8 T + 229 T^{2} + 1288 T^{3} + 20396 T^{4} + 1288 p T^{5} + 229 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.i_iv_bxo_beem |
| 67 | $C_2 \wr S_4$ | \( 1 - 6 T + 57 T^{2} - 572 T^{3} + 9670 T^{4} - 572 p T^{5} + 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ag_cf_awa_ohy |
| 71 | $C_2 \wr S_4$ | \( 1 - 14 T + 268 T^{2} - 2270 T^{3} + 25942 T^{4} - 2270 p T^{5} + 268 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ao_ki_adji_bmju |
| 73 | $C_2 \wr S_4$ | \( 1 + 8 T + 173 T^{2} + 1664 T^{3} + 16372 T^{4} + 1664 p T^{5} + 173 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.i_gr_cma_yfs |
| 79 | $C_2 \wr S_4$ | \( 1 + 26 T + 450 T^{2} + 5138 T^{3} + 50938 T^{4} + 5138 p T^{5} + 450 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ba_ri_hpq_cxje |
| 83 | $C_2 \wr S_4$ | \( 1 + 235 T^{2} - 442 T^{3} + 24806 T^{4} - 442 p T^{5} + 235 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_jb_ara_bksc |
| 89 | $C_2 \wr S_4$ | \( 1 + T + 285 T^{2} + 83 T^{3} + 34780 T^{4} + 83 p T^{5} + 285 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.89.b_kz_df_bzls |
| 97 | $C_2 \wr S_4$ | \( 1 - 3 T + 83 T^{2} - 205 T^{3} + 12976 T^{4} - 205 p T^{5} + 83 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ad_df_ahx_tfc |
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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
−5.44871722011127224815907110526, −5.24609089987041624771889326219, −5.00979219549136520903120233825, −4.84517251868752219802465648158, −4.77054898781748797556803808352, −4.40216556982779276790608045346, −4.35237905364604405378425077745, −4.16792591704755031693736714908, −4.01424148847639429391136010158, −3.98325244687084035745015708113, −3.54465924168303994738816273177, −3.48558844118897320468312645566, −3.38338419481810936570024191881, −3.03172379747920415854515649589, −2.90491801847492373718001721857, −2.84166820180032426761099165597, −2.56132916434468490931651586621, −2.02973023314560363484144554101, −1.88520176970227567244860871378, −1.53519206821855917474501383733, −1.24326405543789894682793432298, −1.06480600938320255846840090115, −0.72524740940632715860339335080, −0.32136672256587507529108765474, −0.23687159845159260808591055317,
0.23687159845159260808591055317, 0.32136672256587507529108765474, 0.72524740940632715860339335080, 1.06480600938320255846840090115, 1.24326405543789894682793432298, 1.53519206821855917474501383733, 1.88520176970227567244860871378, 2.02973023314560363484144554101, 2.56132916434468490931651586621, 2.84166820180032426761099165597, 2.90491801847492373718001721857, 3.03172379747920415854515649589, 3.38338419481810936570024191881, 3.48558844118897320468312645566, 3.54465924168303994738816273177, 3.98325244687084035745015708113, 4.01424148847639429391136010158, 4.16792591704755031693736714908, 4.35237905364604405378425077745, 4.40216556982779276790608045346, 4.77054898781748797556803808352, 4.84517251868752219802465648158, 5.00979219549136520903120233825, 5.24609089987041624771889326219, 5.44871722011127224815907110526
