| L(s) = 1 | − 6·9-s + 20·25-s − 4·29-s − 20·41-s − 28·49-s + 28·53-s + 27·81-s + 44·89-s − 52·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·9-s + 4·25-s − 0.742·29-s − 3.12·41-s − 4·49-s + 3.84·53-s + 3·81-s + 4.66·89-s − 4.89·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724071544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724071544\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.5.a_au_a_fu |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2^3$ | \( 1 + 14 T^{4} + p^{4} T^{8} \) | 4.11.a_a_a_o |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 23 | $C_2^3$ | \( 1 - 994 T^{4} + p^{4} T^{8} \) | 4.23.a_a_a_abmg |
| 29 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.e_i_eu_cvu |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_bc_a_ddm |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.u_hs_csa_uuw |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2^3$ | \( 1 - 1282 T^{4} + p^{4} T^{8} \) | 4.47.a_a_a_abxi |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.abc_pc_aggq_cbgo |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aie_a_bbqk |
| 67 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aem_a_sgs |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_agi_a_zso |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_alg_a_bxzy |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_dwg |
| 89 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.abs_blg_avoe_jdni |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.97.a_aoy_a_dfni |
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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
−6.49695746291395995806772560959, −6.39740734156041367484479555421, −6.35533789700374681779878679686, −5.91663745684891425934127487322, −5.82773742924448125433105682815, −5.32362618354005457784764931618, −5.18188101724575297175271446831, −5.10606867346599081624875237558, −5.01254380231298769648502317228, −4.88279463706779637551110078698, −4.52939095888876984804761808642, −4.14575115895809281609110128980, −3.76127281887857694234866537821, −3.65598813637919085517815423830, −3.47280489036991740611849193263, −3.08523702104777094000804289127, −3.02908369199729169047059251542, −2.69675810148719003658813852852, −2.55978812792098991649621794052, −2.20748240143061753258331766481, −1.72485466471773689071701137038, −1.62878320960489286324611836570, −1.07790855338856607295580208404, −0.70270372243540103495744237347, −0.29728786178881295080083415122,
0.29728786178881295080083415122, 0.70270372243540103495744237347, 1.07790855338856607295580208404, 1.62878320960489286324611836570, 1.72485466471773689071701137038, 2.20748240143061753258331766481, 2.55978812792098991649621794052, 2.69675810148719003658813852852, 3.02908369199729169047059251542, 3.08523702104777094000804289127, 3.47280489036991740611849193263, 3.65598813637919085517815423830, 3.76127281887857694234866537821, 4.14575115895809281609110128980, 4.52939095888876984804761808642, 4.88279463706779637551110078698, 5.01254380231298769648502317228, 5.10606867346599081624875237558, 5.18188101724575297175271446831, 5.32362618354005457784764931618, 5.82773742924448125433105682815, 5.91663745684891425934127487322, 6.35533789700374681779878679686, 6.39740734156041367484479555421, 6.49695746291395995806772560959
