| L(s) = 1 | + 2·3-s + 2·5-s − 9-s − 2·11-s − 4·13-s + 4·15-s − 2·17-s − 6·19-s + 12·23-s − 9·25-s − 2·27-s − 16·29-s + 10·31-s − 4·33-s + 14·37-s − 8·39-s − 20·41-s − 14·43-s − 2·45-s + 20·47-s − 15·49-s − 4·51-s − 4·53-s − 4·55-s − 12·57-s − 2·59-s − 8·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 1/3·9-s − 0.603·11-s − 1.10·13-s + 1.03·15-s − 0.485·17-s − 1.37·19-s + 2.50·23-s − 9/5·25-s − 0.384·27-s − 2.97·29-s + 1.79·31-s − 0.696·33-s + 2.30·37-s − 1.28·39-s − 3.12·41-s − 2.13·43-s − 0.298·45-s + 2.91·47-s − 2.14·49-s − 0.560·51-s − 0.549·53-s − 0.539·55-s − 1.58·57-s − 0.260·59-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.918840632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.918840632\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_f_ak_q |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 13 T^{2} - 22 T^{3} + 84 T^{4} - 22 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ac_n_aw_dg |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 15 T^{2} + 18 T^{3} + 110 T^{4} + 18 p T^{5} + 15 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_p_s_eg |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 34 T^{2} + 58 T^{3} + 522 T^{4} + 58 p T^{5} + 34 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.c_bi_cg_uc |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 29 T^{2} - 46 T^{3} + 304 T^{4} - 46 p T^{5} + 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.c_bd_abu_ls |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 42 T^{2} + 198 T^{3} + 938 T^{4} + 198 p T^{5} + 42 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.g_bq_hq_bkc |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 100 T^{2} - 636 T^{3} + 3414 T^{4} - 636 p T^{5} + 100 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.am_dw_aym_fbi |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 160 T^{2} + 1088 T^{3} + 6414 T^{4} + 1088 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.q_ge_bpw_jms |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 90 T^{2} - 338 T^{3} + 2186 T^{4} - 338 p T^{5} + 90 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ak_dm_ana_dgc |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 14 T + 181 T^{2} - 1334 T^{3} + 9892 T^{4} - 1334 p T^{5} + 181 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ao_gz_abzi_oqm |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 268 T^{2} + 2476 T^{3} + 18246 T^{4} + 2476 p T^{5} + 268 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.u_ki_drg_bazu |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 14 T + 165 T^{2} + 1546 T^{3} + 10928 T^{4} + 1546 p T^{5} + 165 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.o_gj_chm_qei |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 311 T^{2} - 3014 T^{3} + 24622 T^{4} - 3014 p T^{5} + 311 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.au_lz_aely_bkla |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 124 T^{2} + 380 T^{3} + 7398 T^{4} + 380 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.e_eu_oq_kyo |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 206 T^{2} + 314 T^{3} + 17530 T^{4} + 314 p T^{5} + 206 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.c_hy_mc_zyg |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 136 T^{2} - 288 T^{3} + 9726 T^{4} - 288 p T^{5} + 136 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_fg_alc_okc |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 198 T^{2} - 1110 T^{3} + 17546 T^{4} - 1110 p T^{5} + 198 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ag_hq_abqs_zyw |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 28 T + 551 T^{2} - 6994 T^{3} + 69526 T^{4} - 6994 p T^{5} + 551 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.abc_vf_akja_dywc |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 156 T^{2} - 1144 T^{3} + 17702 T^{4} - 1144 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ai_ga_absa_baew |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 312 T^{2} - 3680 T^{3} + 36782 T^{4} - 3680 p T^{5} + 312 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.aq_ma_aflo_ccks |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 22 T + 430 T^{2} + 5366 T^{3} + 56730 T^{4} + 5366 p T^{5} + 430 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.w_qo_hyk_dfxy |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 328 T^{2} - 2104 T^{3} + 42606 T^{4} - 2104 p T^{5} + 328 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ai_mq_adcy_clas |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 480 T^{2} + 5932 T^{3} + 74270 T^{4} + 5932 p T^{5} + 480 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.u_sm_iue_efwo |
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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
−6.21320786105030226543416876647, −5.69729971864101764787739724143, −5.67304292836233072536531113600, −5.48669017489256770625374761810, −5.44193314232044428782733172409, −5.09324891599780543314009432418, −4.79208249579375619736410792292, −4.75050651104263734229777272995, −4.59864761156340375453614749156, −4.30880157005063596646692833265, −3.86187365827992859728312431592, −3.77286505967946513451235549468, −3.69161508714055661651281305082, −3.21237430602130676329633906745, −3.09252862735841171932293609217, −2.90472076701958770569679140531, −2.72966136342155465569331583106, −2.27411434804865918463188622815, −2.18080504455546873946286372696, −2.13299795313655238726639179675, −1.81060158180395828277235196006, −1.50429973040157000388156488377, −1.11244551635409229712132269179, −0.43090262408418840311393223561, −0.39047313863323581865825712840,
0.39047313863323581865825712840, 0.43090262408418840311393223561, 1.11244551635409229712132269179, 1.50429973040157000388156488377, 1.81060158180395828277235196006, 2.13299795313655238726639179675, 2.18080504455546873946286372696, 2.27411434804865918463188622815, 2.72966136342155465569331583106, 2.90472076701958770569679140531, 3.09252862735841171932293609217, 3.21237430602130676329633906745, 3.69161508714055661651281305082, 3.77286505967946513451235549468, 3.86187365827992859728312431592, 4.30880157005063596646692833265, 4.59864761156340375453614749156, 4.75050651104263734229777272995, 4.79208249579375619736410792292, 5.09324891599780543314009432418, 5.44193314232044428782733172409, 5.48669017489256770625374761810, 5.67304292836233072536531113600, 5.69729971864101764787739724143, 6.21320786105030226543416876647
