| L(s) = 1 | + 3·4-s + 2·9-s + 5·16-s + 16·31-s + 6·36-s − 20·41-s − 4·49-s + 3·64-s + 32·71-s − 16·79-s − 15·81-s + 4·89-s + 30·121-s + 48·124-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s + 163-s − 60·164-s + 167-s − 52·169-s + 173-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2/3·9-s + 5/4·16-s + 2.87·31-s + 36-s − 3.12·41-s − 4/7·49-s + 3/8·64-s + 3.79·71-s − 1.80·79-s − 5/3·81-s + 0.423·89-s + 2.72·121-s + 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 4.68·164-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481595648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481595648\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) | |
| 5 | | \( 1 \) | |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ac_a_t |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_e_a_dy |
| 11 | $C_2^2$ | \( ( 1 - 15 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abe_a_rz |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.13.a_ca_a_bna |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_aby_a_buh |
| 19 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ack_a_cmt |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_aci_a_cxi |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.31.aq_im_acpc_rso |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_acy_a_gew |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.41.u_mc_ejw_bibb |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_em_a_klq |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_aci_a_hwo |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_am_a_ijm |
| 59 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agy_a_whi |
| 61 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_au_a_lec |
| 67 | $C_2^2$ | \( ( 1 + 71 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_fm_a_utf |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.71.abg_zs_ancy_fbmc |
| 73 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ahm_a_bdrv |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.79.q_pw_fzs_dafm |
| 83 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_hy_a_bkbz |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.89.ae_ny_abpg_ctxb |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_aoq_a_ddgg |
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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
−9.190607393385172179734590056136, −8.605537402077930067745739153173, −8.586212018657907730774990811163, −8.375610926513820072529665094237, −8.001034426001162320652754232316, −7.73565745543270063342162743222, −7.56656523134970228466926862886, −7.08350979943628006797820629194, −6.85313421774544453589408028956, −6.63224920823709166006098030981, −6.58208622324050151875550052676, −6.22374337506708269031862472825, −5.85502028149049305838723519637, −5.37092937476263134399023166095, −5.33017780950534025516329791163, −4.68010370972612999007388483110, −4.59533903115521122854606799298, −4.21054357404571717333386696193, −3.65581506697939024858887814472, −3.19503564901073142508403368303, −3.14325443589533872306967242419, −2.51054797241481095901075845225, −2.13070617041045605093388771132, −1.67095650089860251827778479311, −1.06737227812559465230353803346,
1.06737227812559465230353803346, 1.67095650089860251827778479311, 2.13070617041045605093388771132, 2.51054797241481095901075845225, 3.14325443589533872306967242419, 3.19503564901073142508403368303, 3.65581506697939024858887814472, 4.21054357404571717333386696193, 4.59533903115521122854606799298, 4.68010370972612999007388483110, 5.33017780950534025516329791163, 5.37092937476263134399023166095, 5.85502028149049305838723519637, 6.22374337506708269031862472825, 6.58208622324050151875550052676, 6.63224920823709166006098030981, 6.85313421774544453589408028956, 7.08350979943628006797820629194, 7.56656523134970228466926862886, 7.73565745543270063342162743222, 8.001034426001162320652754232316, 8.375610926513820072529665094237, 8.586212018657907730774990811163, 8.605537402077930067745739153173, 9.190607393385172179734590056136
