| L(s) = 1 | − 4·3-s − 4·5-s + 8·9-s + 16·15-s + 12·23-s + 2·25-s − 20·27-s − 16·31-s + 20·37-s − 32·45-s + 20·47-s − 12·53-s − 12·67-s − 48·69-s − 16·71-s − 8·75-s + 50·81-s + 64·93-s + 20·97-s − 36·103-s − 80·111-s + 4·113-s − 48·115-s − 22·121-s + 28·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s + 8/3·9-s + 4.13·15-s + 2.50·23-s + 2/5·25-s − 3.84·27-s − 2.87·31-s + 3.28·37-s − 4.77·45-s + 2.91·47-s − 1.64·53-s − 1.46·67-s − 5.77·69-s − 1.89·71-s − 0.923·75-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 − 4.47·115-s − 2·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2051507706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2051507706\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | |
| 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.e_i_u_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.am_cu_asy_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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.37.au_hs_acoy_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.au_hs_acwq_ydq |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.m_cu_bgu_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.m_cu_bng_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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.97.au_hs_aejc_cigc |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.080618950628470517460730948177, −8.953867326704648180682716666181, −8.206221519761368504079860496004, −8.145332693899268681450911783029, −7.80572327821513110434623417221, −7.40896486788317958859839849674, −7.35712120095770484475281893937, −7.20416048894738857540437318043, −7.03035059618103168521804987436, −6.33727932095723965907224023997, −6.12681930079027404378820652672, −6.02690240422081118497000717493, −5.65278038763268869841334979135, −5.31850844865269093506016678717, −5.25756665434140280463819636959, −4.69487706068472971619198575535, −4.42257670886515236174433347081, −4.14074196460287074107750495619, −3.90319275988916905886317694198, −3.55806256093908403167349226159, −3.00368105790245311833605926492, −2.62926123939872058810029953376, −1.82548392297199413169296968148, −1.14899670544842882223404103013, −0.36031884962829309962385165188,
0.36031884962829309962385165188, 1.14899670544842882223404103013, 1.82548392297199413169296968148, 2.62926123939872058810029953376, 3.00368105790245311833605926492, 3.55806256093908403167349226159, 3.90319275988916905886317694198, 4.14074196460287074107750495619, 4.42257670886515236174433347081, 4.69487706068472971619198575535, 5.25756665434140280463819636959, 5.31850844865269093506016678717, 5.65278038763268869841334979135, 6.02690240422081118497000717493, 6.12681930079027404378820652672, 6.33727932095723965907224023997, 7.03035059618103168521804987436, 7.20416048894738857540437318043, 7.35712120095770484475281893937, 7.40896486788317958859839849674, 7.80572327821513110434623417221, 8.145332693899268681450911783029, 8.206221519761368504079860496004, 8.953867326704648180682716666181, 9.080618950628470517460730948177
