| L(s) = 1 | − 3-s − 2·5-s − 3·7-s − 3·13-s + 2·15-s + 11·17-s + 11·19-s + 3·21-s − 20·23-s + 5·25-s − 2·29-s − 7·31-s + 6·35-s − 5·37-s + 3·39-s + 17·41-s − 28·43-s − 22·47-s − 3·49-s − 11·51-s − 3·53-s − 11·57-s − 27·59-s + 6·65-s − 22·67-s + 20·69-s − 18·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.13·7-s − 0.832·13-s + 0.516·15-s + 2.66·17-s + 2.52·19-s + 0.654·21-s − 4.17·23-s + 25-s − 0.371·29-s − 1.25·31-s + 1.01·35-s − 0.821·37-s + 0.480·39-s + 2.65·41-s − 4.26·43-s − 3.20·47-s − 3/7·49-s − 1.54·51-s − 0.412·53-s − 1.45·57-s − 3.51·59-s + 0.744·65-s − 2.68·67-s + 2.40·69-s − 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7222295510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7222295510\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) | |
| 11 | | \( 1 \) | |
| good | 5 | $C_2^2:C_4$ | \( 1 + 2 T - T^{2} + 8 T^{3} + 41 T^{4} + 8 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.c_ab_i_bp |
| 7 | $C_2^2:C_4$ | \( 1 + 3 T + 12 T^{2} + 5 p T^{3} + 141 T^{4} + 5 p^{2} T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.d_m_bj_fl |
| 13 | $C_2^2:C_4$ | \( 1 + 3 T + 6 T^{2} + 59 T^{3} + 339 T^{4} + 59 p T^{5} + 6 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.d_g_ch_nb |
| 17 | $C_2^2:C_4$ | \( 1 - 11 T + 29 T^{2} + 123 T^{3} - 1036 T^{4} + 123 p T^{5} + 29 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.al_bd_et_abnw |
| 19 | $C_4\times C_2$ | \( 1 - 11 T + 27 T^{2} + 137 T^{3} - 1120 T^{4} + 137 p T^{5} + 27 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.al_bb_fh_abrc |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.23.u_ji_cui_pvn |
| 29 | $C_4\times C_2$ | \( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.c_az_aee_tp |
| 31 | $C_2^2:C_4$ | \( 1 + 7 T - 7 T^{2} - 241 T^{3} - 1220 T^{4} - 241 p T^{5} - 7 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.h_ah_ajh_abuy |
| 37 | $C_4\times C_2$ | \( 1 + 5 T - 22 T^{2} + 25 T^{3} + 1579 T^{4} + 25 p T^{5} - 22 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.f_aw_z_cit |
| 41 | $C_2^2:C_4$ | \( 1 - 17 T + 98 T^{2} - 9 p T^{3} + 55 p T^{4} - 9 p^{2} T^{5} + 98 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ar_du_aof_dit |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.bc_qk_goe_bxzf |
| 47 | $C_2^2:C_4$ | \( 1 + 22 T + 257 T^{2} + 2370 T^{3} + 18191 T^{4} + 2370 p T^{5} + 257 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.w_jx_dne_baxr |
| 53 | $C_2^2:C_4$ | \( 1 + 3 T - 19 T^{2} - 351 T^{3} + 364 T^{4} - 351 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.d_at_ann_oa |
| 59 | $C_2^2:C_4$ | \( 1 + 27 T + 395 T^{2} + 4287 T^{3} + 36784 T^{4} + 4287 p T^{5} + 395 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.bb_pf_gix_ccku |
| 61 | $C_2^2:C_4$ | \( 1 + 29 T^{2} + 240 T^{3} + 3001 T^{4} + 240 p T^{5} + 29 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_bd_jg_ell |
| 67 | $D_{4}$ | \( ( 1 + 11 T + 163 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.w_rf_hmq_cyov |
| 71 | $C_2^2:C_4$ | \( 1 + 18 T + 253 T^{2} + 2826 T^{3} + 28855 T^{4} + 2826 p T^{5} + 253 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.s_jt_ees_bqrv |
| 73 | $C_2^2:C_4$ | \( 1 - 24 T + 143 T^{2} + 1530 T^{3} - 27539 T^{4} + 1530 p T^{5} + 143 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ay_fn_cgw_abotf |
| 79 | $C_2^2:C_4$ | \( 1 + 13 T + 60 T^{2} + 1013 T^{3} + 15269 T^{4} + 1013 p T^{5} + 60 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.n_ci_bmz_wph |
| 83 | $C_2^2:C_4$ | \( 1 - 9 T - 52 T^{2} + 675 T^{3} + 121 T^{4} + 675 p T^{5} - 52 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.aj_aca_zz_er |
| 89 | $D_{4}$ | \( ( 1 - 12 T + p T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_mk_agii_cvbv |
| 97 | $C_2^2:C_4$ | \( 1 - 32 T + 537 T^{2} - 7060 T^{3} + 77501 T^{4} - 7060 p T^{5} + 537 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.abg_ur_aklo_ekqv |
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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
−6.85131755728321752674000220137, −6.38878324529626519270906793471, −6.36279998000892415689347753562, −6.01191543384556252002799727183, −5.97358153957744363089138093412, −5.78216591640640277040195641864, −5.72311123336606262499288480156, −4.95633798636671405790415606309, −4.95574486547790861617603144975, −4.93137238187516769041975225843, −4.84267280057385079192819758916, −4.44083935580810520964428762334, −3.80819634138587555955935870302, −3.73640379420835171791640954100, −3.54843465667293526480641115698, −3.36818860948670918735429698487, −3.19587672567845539837346410591, −2.99806328541725908632221239801, −2.76007104722348447627233631835, −1.92717147076232389258287799645, −1.86723339857794140112020239627, −1.63170575166913840041972344877, −1.28372697875000684321237761479, −0.42329324245763112468568808376, −0.35121800442816339555797195441,
0.35121800442816339555797195441, 0.42329324245763112468568808376, 1.28372697875000684321237761479, 1.63170575166913840041972344877, 1.86723339857794140112020239627, 1.92717147076232389258287799645, 2.76007104722348447627233631835, 2.99806328541725908632221239801, 3.19587672567845539837346410591, 3.36818860948670918735429698487, 3.54843465667293526480641115698, 3.73640379420835171791640954100, 3.80819634138587555955935870302, 4.44083935580810520964428762334, 4.84267280057385079192819758916, 4.93137238187516769041975225843, 4.95574486547790861617603144975, 4.95633798636671405790415606309, 5.72311123336606262499288480156, 5.78216591640640277040195641864, 5.97358153957744363089138093412, 6.01191543384556252002799727183, 6.36279998000892415689347753562, 6.38878324529626519270906793471, 6.85131755728321752674000220137
