| L(s) = 1 | + 4·3-s + 8·9-s + 12·13-s − 16-s − 12·17-s − 12·19-s − 12·23-s − 8·25-s + 16·27-s + 12·29-s − 12·31-s − 8·37-s + 48·39-s − 12·43-s + 12·47-s − 4·48-s − 48·51-s + 24·53-s − 48·57-s − 16·67-s − 48·69-s + 12·71-s + 24·73-s − 32·75-s + 24·79-s + 33·81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 3.32·13-s − 1/4·16-s − 2.91·17-s − 2.75·19-s − 2.50·23-s − 8/5·25-s + 3.07·27-s + 2.22·29-s − 2.15·31-s − 1.31·37-s + 7.68·39-s − 1.82·43-s + 1.75·47-s − 0.577·48-s − 6.72·51-s + 3.29·53-s − 6.35·57-s − 1.95·67-s − 5.77·69-s + 1.42·71-s + 2.80·73-s − 3.69·75-s + 2.70·79-s + 11/3·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.044786948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044786948\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) | |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) | |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 31 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ae_i_aq_bf |
| 7 | $C_2^3$ | \( 1 - 73 T^{4} + p^{4} T^{8} \) | 4.7.a_a_a_acv |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.am_cu_aoi_cjm |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 312 T^{3} + 1271 T^{4} + 312 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.m_cu_ma_bwx |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.19.m_fa_bem_gjj |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.m_cu_lo_bvy |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.am_fe_abka_jdj |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.m_fm_bmu_khv |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 288 T^{3} + 2591 T^{4} + 288 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.i_bg_lc_dvr |
| 41 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1554 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_ace_a_chu |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 926 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.m_cu_lo_bjq |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 732 T^{3} + 7246 T^{4} - 732 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.am_cu_abce_kss |
| 53 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2400 T^{3} + 17791 T^{4} - 2400 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ay_lc_adoi_baih |
| 59 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aiy_a_befi |
| 61 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 15819 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_ahi_a_xkl |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.q_ey_ciy_bbmo |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.am_lq_adqe_bwqt |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ay_lc_afdw_ccny |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ay_tc_ajbk_dtyo |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 780 T^{3} + 8126 T^{4} + 780 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.m_cu_bea_mao |
| 89 | $D_4\times C_2$ | \( 1 - 194 T^{2} + 22659 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ahm_a_bhnn |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 552 T^{3} + 8738 T^{4} - 552 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ai_bg_avg_myc |
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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
−10.36981264734809617538451475191, −9.765061844068366462901407380717, −9.128431095279566032597316304171, −9.059519036598965859776633533989, −8.808774128457611489355282898851, −8.779940724551212268075874414571, −8.422900226516719833074242586972, −8.234419204157333859961660367458, −8.129987811182682202157461426846, −7.67611878095049031690744826134, −7.20494011903028957006323110608, −6.66982817656714865729393670030, −6.38135762585198493384827002430, −6.37611975826394405004313110237, −6.28439109269128197840656246327, −5.41647617266707158627460822550, −5.14466741477393392422053021911, −4.24691816132634715133386340351, −4.07323826733133829954506404219, −3.89226970392756088836233050041, −3.85074730143178893186960108990, −3.09814769631830291355157683830, −2.35787865031481513039117495277, −2.00281678291443671113746900445, −1.96414213407946840078860083951,
1.96414213407946840078860083951, 2.00281678291443671113746900445, 2.35787865031481513039117495277, 3.09814769631830291355157683830, 3.85074730143178893186960108990, 3.89226970392756088836233050041, 4.07323826733133829954506404219, 4.24691816132634715133386340351, 5.14466741477393392422053021911, 5.41647617266707158627460822550, 6.28439109269128197840656246327, 6.37611975826394405004313110237, 6.38135762585198493384827002430, 6.66982817656714865729393670030, 7.20494011903028957006323110608, 7.67611878095049031690744826134, 8.129987811182682202157461426846, 8.234419204157333859961660367458, 8.422900226516719833074242586972, 8.779940724551212268075874414571, 8.808774128457611489355282898851, 9.059519036598965859776633533989, 9.128431095279566032597316304171, 9.765061844068366462901407380717, 10.36981264734809617538451475191
