| L(s) = 1 | + 4·3-s − 2·5-s − 8·7-s + 13·9-s + 6·13-s − 8·15-s + 6·17-s + 8·19-s − 32·21-s + 8·23-s + 5·25-s + 30·27-s + 10·29-s + 10·31-s + 16·35-s + 6·37-s + 24·39-s + 10·41-s − 26·45-s + 37·49-s + 24·51-s − 14·53-s + 32·57-s − 8·59-s + 6·61-s − 104·63-s − 12·65-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s − 3.02·7-s + 13/3·9-s + 1.66·13-s − 2.06·15-s + 1.45·17-s + 1.83·19-s − 6.98·21-s + 1.66·23-s + 25-s + 5.77·27-s + 1.85·29-s + 1.79·31-s + 2.70·35-s + 0.986·37-s + 3.84·39-s + 1.56·41-s − 3.87·45-s + 37/7·49-s + 3.36·51-s − 1.92·53-s + 4.23·57-s − 1.04·59-s + 0.768·61-s − 13.1·63-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.103064845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.103064845\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | | \( 1 \) | |
| good | 3 | $C_2^2:C_4$ | \( 1 - 4 T + p T^{2} + 10 T^{3} - 29 T^{4} + 10 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ae_d_k_abd |
| 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 | $C_2^2:C_4$ | \( 1 + 8 T + 27 T^{2} + 10 p T^{3} + 191 T^{4} + 10 p^{2} T^{5} + 27 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.i_bb_cs_hj |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 3 T^{2} + 80 T^{3} - 399 T^{4} + 80 p T^{5} + 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ag_d_dc_apj |
| 17 | $C_2^2:C_4$ | \( 1 - 6 T - T^{2} + 108 T^{3} - 491 T^{4} + 108 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_ab_ee_asx |
| 19 | $C_2^2:C_4$ | \( 1 - 8 T + 15 T^{2} - 58 T^{3} + 539 T^{4} - 58 p T^{5} + 15 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ai_p_acg_ut |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ai_cy_aqi_dzq |
| 29 | $C_2^2:C_4$ | \( 1 - 10 T + 31 T^{2} + 160 T^{3} - 2079 T^{4} + 160 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ak_bf_ge_adbz |
| 31 | $C_2^2:C_4$ | \( 1 - 10 T + 9 T^{2} + 130 T^{3} - 409 T^{4} + 130 p T^{5} + 9 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ak_j_fa_apt |
| 37 | $C_2^2:C_4$ | \( 1 - 6 T - T^{2} - 192 T^{3} + 2449 T^{4} - 192 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ag_ab_ahk_dqf |
| 41 | $C_2^2:C_4$ | \( 1 - 10 T + 19 T^{2} + 340 T^{3} - 3699 T^{4} + 340 p T^{5} + 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ak_t_nc_afmh |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2^2:C_4$ | \( 1 - 37 T^{2} - 210 T^{3} + 1999 T^{4} - 210 p T^{5} - 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abl_aic_cyx |
| 53 | $C_2^2:C_4$ | \( 1 + 14 T + 43 T^{2} + 160 T^{3} + 2961 T^{4} + 160 p T^{5} + 43 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.o_br_ge_ejx |
| 59 | $C_2^2:C_4$ | \( 1 + 8 T + 55 T^{2} + 718 T^{3} + 8499 T^{4} + 718 p T^{5} + 55 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.i_cd_bbq_mox |
| 61 | $C_2^2:C_4$ | \( 1 - 6 T - 45 T^{2} + 416 T^{3} + 609 T^{4} + 416 p T^{5} - 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ag_abt_qa_xl |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.aq_hw_adgi_bhfq |
| 71 | $C_2^2:C_4$ | \( 1 - 10 T - 11 T^{2} + 790 T^{3} - 6489 T^{4} + 790 p T^{5} - 11 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ak_al_bek_ajpp |
| 73 | $C_2^2:C_4$ | \( 1 - 6 T + 63 T^{2} - 740 T^{3} + 10341 T^{4} - 740 p T^{5} + 63 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ag_cl_abcm_pht |
| 79 | $C_2^2:C_4$ | \( 1 + 18 T + 65 T^{2} - 1362 T^{3} - 21641 T^{4} - 1362 p T^{5} + 65 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.s_cn_acak_abgaj |
| 83 | $C_2^2:C_4$ | \( 1 + 34 T + 553 T^{2} + 6230 T^{3} + 59451 T^{4} + 6230 p T^{5} + 553 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.bi_vh_jfq_djyp |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.q_pw_gfw_dhby |
| 97 | $C_2^2:C_4$ | \( 1 + 18 T + 227 T^{2} + 3060 T^{3} + 39541 T^{4} + 3060 p T^{5} + 227 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.s_it_ens_cgmv |
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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
−7.937261748050835257795376719047, −7.921839358720243012940176755082, −7.34728901870440665801905508973, −7.23667885501764901745927886175, −7.14127905763164215739289470271, −6.66623035760577366553208148121, −6.61594517305613766809488608179, −6.47374851336202765233537204903, −6.17518770396525502728439652262, −5.64862261427245014913852925531, −5.54231661563834750621789974024, −5.14247071291650841439004644711, −4.75742052953763658760168065716, −4.18480666420232805795138244021, −4.16397685369976512602406170840, −4.05283133593475884171647846471, −3.58037246900776673949149662476, −3.18126329673834939276511521875, −3.16622664921507288412484857846, −2.99799713131858320269564474451, −2.61576590852198719548922799088, −2.57122579690891634744221049847, −1.25393392761175148878145354219, −1.24518582605777636966553691239, −0.932517388457358335170973452081,
0.932517388457358335170973452081, 1.24518582605777636966553691239, 1.25393392761175148878145354219, 2.57122579690891634744221049847, 2.61576590852198719548922799088, 2.99799713131858320269564474451, 3.16622664921507288412484857846, 3.18126329673834939276511521875, 3.58037246900776673949149662476, 4.05283133593475884171647846471, 4.16397685369976512602406170840, 4.18480666420232805795138244021, 4.75742052953763658760168065716, 5.14247071291650841439004644711, 5.54231661563834750621789974024, 5.64862261427245014913852925531, 6.17518770396525502728439652262, 6.47374851336202765233537204903, 6.61594517305613766809488608179, 6.66623035760577366553208148121, 7.14127905763164215739289470271, 7.23667885501764901745927886175, 7.34728901870440665801905508973, 7.921839358720243012940176755082, 7.937261748050835257795376719047
