| L(s) = 1 | + 4·3-s − 8·5-s + 8·9-s − 4·11-s − 32·15-s + 4·23-s + 38·25-s + 20·27-s − 8·31-s − 16·33-s + 12·37-s − 64·45-s − 12·47-s − 4·53-s + 32·55-s − 12·67-s + 16·69-s + 32·71-s + 152·75-s + 50·81-s − 32·93-s − 28·97-s − 32·99-s − 36·103-s + 48·111-s + 36·113-s − 32·115-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.57·5-s + 8/3·9-s − 1.20·11-s − 8.26·15-s + 0.834·23-s + 38/5·25-s + 3.84·27-s − 1.43·31-s − 2.78·33-s + 1.97·37-s − 9.54·45-s − 1.75·47-s − 0.549·53-s + 4.31·55-s − 1.46·67-s + 1.92·69-s + 3.79·71-s + 17.5·75-s + 50/9·81-s − 3.31·93-s − 2.84·97-s − 3.21·99-s − 3.54·103-s + 4.55·111-s + 3.38·113-s − 2.98·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9358538894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9358538894\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | |
| 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 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.7.a_a_a_du |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) | 4.13.a_a_a_alq |
| 17 | $C_2^3$ | \( 1 - 382 T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_aos |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ae_a_bby |
| 23 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_i_adw_bvy |
| 29 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_bk_a_cze |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.31.i_fs_bdw_ktq |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.am_cu_azk_img |
| 41 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_adg_a_hpe |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.43.a_a_a_fmg |
| 47 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.m_cu_bea_mao |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.e_i_im_iyo |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agi_a_ugk |
| 61 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agi_a_uyw |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.71.abg_zs_ancy_fbmc |
| 73 | $C_2^3$ | \( 1 + 5218 T^{4} + p^{4} T^{8} \) | 4.73.a_a_a_hss |
| 79 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_jc_a_bnbq |
| 83 | $C_2^3$ | \( 1 - 6382 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_ajlm |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aky_a_cbgw |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.bc_pc_ica_duhq |
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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
−7.38390923781100613247106850389, −7.23681923576881146176516307554, −6.97296079755030419946263333063, −6.90507118741894181018553222206, −6.49458654643036663563020579658, −6.42203631009995882523089360681, −5.84411397990092720221205971094, −5.63329052204039750159589557213, −5.16833593859894865801655382935, −4.98475077094329710601840110633, −4.68463650320301155002658652140, −4.68284539139480448996594629466, −4.25795504620391457954661733260, −4.13243547141818630032685068060, −3.69971638903882813429817144514, −3.57152674002739626985366123614, −3.42008358040490609232441930707, −3.05644179318958244371175155686, −2.83870363636984181184969177419, −2.77186646851212872030294956847, −2.33406341755051100167471793626, −2.00303078980514519127025773793, −1.10881640068334174788748917847, −1.10770838453874340512936680057, −0.23017278219018337876147565176,
0.23017278219018337876147565176, 1.10770838453874340512936680057, 1.10881640068334174788748917847, 2.00303078980514519127025773793, 2.33406341755051100167471793626, 2.77186646851212872030294956847, 2.83870363636984181184969177419, 3.05644179318958244371175155686, 3.42008358040490609232441930707, 3.57152674002739626985366123614, 3.69971638903882813429817144514, 4.13243547141818630032685068060, 4.25795504620391457954661733260, 4.68284539139480448996594629466, 4.68463650320301155002658652140, 4.98475077094329710601840110633, 5.16833593859894865801655382935, 5.63329052204039750159589557213, 5.84411397990092720221205971094, 6.42203631009995882523089360681, 6.49458654643036663563020579658, 6.90507118741894181018553222206, 6.97296079755030419946263333063, 7.23681923576881146176516307554, 7.38390923781100613247106850389
