| L(s) = 1 | + 4·3-s + 8·9-s − 12·23-s + 20·27-s − 16·31-s − 20·37-s − 20·47-s + 12·53-s + 12·67-s − 48·69-s − 16·71-s + 50·81-s − 64·93-s − 20·97-s + 36·103-s − 80·111-s − 4·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 80·141-s + 149-s + 151-s + 157-s + 48·159-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 2.50·23-s + 3.84·27-s − 2.87·31-s − 3.28·37-s − 2.91·47-s + 1.64·53-s + 1.46·67-s − 5.77·69-s − 1.89·71-s + 50/9·81-s − 6.63·93-s − 2.03·97-s + 3.54·103-s − 7.59·111-s − 0.376·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.73·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 3.80·159-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5251859727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5251859727\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.ae_i_au_bu |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) | 4.7.a_a_a_abi |
| 13 | $C_2^3$ | \( 1 - 322 T^{4} + p^{4} T^{8} \) | 4.13.a_a_a_amk |
| 17 | $C_2^3$ | \( 1 - 434 T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_aqs |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.19.a_cy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.m_cu_sy_emw |
| 29 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_bc_a_cug |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.31.q_im_cpc_rso |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.37.u_hs_coy_ssc |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2^3$ | \( 1 + 398 T^{4} + p^{4} T^{8} \) | 4.43.a_a_a_pi |
| 47 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.u_hs_cwq_ydq |
| 53 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.am_cu_abgu_oli |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abk_a_kug |
| 61 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ee_a_pik |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.am_cu_abng_uxi |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.71.q_oq_fky_cnhm |
| 73 | $C_2^3$ | \( 1 + 4718 T^{4} + p^{4} T^{8} \) | 4.73.a_a_a_gzm |
| 79 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_abk_a_syo |
| 83 | $C_2^3$ | \( 1 + 6958 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_khq |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_acq_a_zdu |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) | 4.97.u_hs_ejc_cigc |
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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
−6.93097659270782873510646729729, −6.92443483625134747948605020464, −6.87859685570220996173127656087, −6.49954160964974157061061271287, −6.25703445908689658354984548570, −5.85881157335167865238368594188, −5.70689003166310928945395428664, −5.41549153723501915197444201926, −5.32748944776425961089887015161, −4.84436734129907640698705987305, −4.72215532312563727067919901051, −4.51340150887006159365340450850, −4.12896102099316064028493988756, −3.71296577050104981333802268548, −3.60673565925220604933252874432, −3.56998519319244199689659931933, −3.37193149075343909793007263338, −2.82585600300257893474795111264, −2.80441882781253588254289671506, −2.16130833678251663393415989637, −2.04978035851968256030555848026, −2.03568262591685441099556083814, −1.35207419724941572119890959015, −1.32950219168297337010734421521, −0.11188106919991573900149237059,
0.11188106919991573900149237059, 1.32950219168297337010734421521, 1.35207419724941572119890959015, 2.03568262591685441099556083814, 2.04978035851968256030555848026, 2.16130833678251663393415989637, 2.80441882781253588254289671506, 2.82585600300257893474795111264, 3.37193149075343909793007263338, 3.56998519319244199689659931933, 3.60673565925220604933252874432, 3.71296577050104981333802268548, 4.12896102099316064028493988756, 4.51340150887006159365340450850, 4.72215532312563727067919901051, 4.84436734129907640698705987305, 5.32748944776425961089887015161, 5.41549153723501915197444201926, 5.70689003166310928945395428664, 5.85881157335167865238368594188, 6.25703445908689658354984548570, 6.49954160964974157061061271287, 6.87859685570220996173127656087, 6.92443483625134747948605020464, 6.93097659270782873510646729729
